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Question ID 3008cfc3 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in...

Question ID 3008cfc3 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 3008cfc3 The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is , where and are constants. What is the value of ? ID: 3008cfc3 Answer Correct Answer: 33 Rationale The correct answer is 33. It’s given in the table that the coordinates of two points on a line in the xy-plane are ( 𝑘, 13 ) and ( 𝑘 + 7, - 15 ). The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, 𝑚, between any two points, 𝑥1 , 𝑦1 and 𝑥2 , 𝑦2 , on the line can be calculated using the slope 𝑦 -𝑦 formula, 𝑚 = 2 1. It follows that the slope of the line with the given points from the table, ( 𝑘, 13 ) and ( 𝑘 + 7, - 15 ) , is 𝑥2 - 𝑥1 𝑚 = 𝑘-15+ 7- 13- 𝑘 , which is equivalent to 𝑚 = -28 7 , or 𝑚 = - 4. It's given that the y-intercept of the line is ( 𝑘 - 5, 𝑏 ). Substituting -4 for 𝑚 and the coordinates of the points ( 𝑘 - 5, 𝑏 ) and ( 𝑘, 13 ) into the slope formula yields -4 = 𝑘13- 𝑘- -𝑏5 , which is equivalent to -4 = 𝑘 13- 𝑘-+𝑏 5 , or -4 = 135- 𝑏. Multiplying both sides of this equation by 5 yields -20 = 13 - 𝑏. Subtracting 13 from both sides of this equation yields -33 = - 𝑏. Dividing both sides of this equation by -1 yields 𝑏 = 33. Therefore, the value of 𝑏 is 33. Question Difficulty: Hard Question ID d1b66ae6 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: d1b66ae6 If satisfies the system of equations above, what is the value of y ? ID: d1b66ae6 Answer Rationale The correct answer is. One method for solving the system of equations for y is to add corresponding sides of the two equations. Adding the left-hand sides gives , or 4y. Adding the right-hand sides yields. It follows that. Finally, dividing both sides of by 4 yields or. Note that 3/2 and 1.5 are examples of ways to enter a correct answer. Question Difficulty: Hard Question ID 3cdbf026 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 3cdbf026 The graph of the equation is a line in the xy-plane, where a and k are constants. If the line contains the points and , what is the value of k? A. B. C. 2 D. 3 ID: 3cdbf026 Answer Correct Answer: A Rationale Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation, , where m is the slope and b is the y-coordinate of the y-intercept. The equation can be rewritten in the form. One of the given points, , is the y-intercept. Thus, the y-coordinate of the y-intercept must be equal to. Multiplying both sides by k gives. Dividing both sides by gives. Choices B, C, and D are incorrect and may result from errors made rewriting the given equation. Question Difficulty: Hard Question ID ff501705 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: ff501705 In the given system of equations, is a constant. If the system has no solution, what is the value of ? ID: ff501705 Answer Correct Answer: 6 Rationale The correct answer is 6. A system of two linear equations in two variables, 𝑥 and 𝑦, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, 𝐴𝑥 + 𝐵𝑦 = 𝐶 and 𝐷𝑥 + 𝐸𝑦 = 𝐹, are parallel if the coefficients for 𝑥 and 𝑦 in one equation are proportional to the corresponding coefficients in 𝐷 𝐸 𝐹 the other equation, meaning = ; and the lines are distinct if the constants are not proportional, meaning is not equal to 𝐴 𝐵 𝐶 𝐷 𝐸 3 1 𝐴 or. The first equation in the given system is 𝑦 - 𝑥 𝐵 2 4 = 23 - 32 𝑦. Multiplying each side of this equation by 12 yields 18𝑦 - 3𝑥 = 8 - 18𝑦. Adding 18𝑦 to each side of this equation yields 36𝑦 - 3𝑥 = 8, or -3𝑥 + 36𝑦 = 8. The second equation in the 1 3 9 given system is 𝑥 + = 𝑝𝑦 +. Multiplying each side of this equation by 2 yields 𝑥 + 3 = 2𝑝𝑦 + 9. Subtracting 2𝑝𝑦 from each 2 2 2 side of this equation yields 𝑥 + 3 - 2𝑝𝑦 = 9. Subtracting 3 from each side of this equation yields 𝑥 - 2𝑝𝑦 = 6. Therefore, the two equations in the given system, written in standard form, are -3𝑥 + 36𝑦 = 8 and 𝑥 - 2𝑝𝑦 = 6. As previously stated, if this system has no solution, the lines represented by the equations in the xy-plane are parallel and distinct, meaning the 1 -2𝑝 1 𝑝 6 1 6 1 proportion = , or - = - , is true and the proportion = is not true. The proportion = is not true. Multiplying each -3 36 3 18 8 -3 8 -3 1 side of the true proportion, - 3 = - 18𝑝 , by -18 yields 6 = 𝑝. Therefore, if the system has no solution, then the value of 𝑝 is 6. Question Difficulty: Hard Question ID 8c5e6702 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: 8c5e6702 A window repair specialist charges for the first two hours of repair plus an hourly fee for each additional hour. The total cost for hours of repair is. Which function gives the total cost, in dollars, for hours of repair, where ? A. B. C. D. ID: 8c5e6702 Answer Correct Answer: A Rationale Choice A is correct. It’s given that the window repair specialist charges $ 220 for the first two hours of repair plus an hourly fee for each additional hour. Let 𝑛 represent the hourly fee for each additional hour after the first two hours. Since it’s given that 𝑥 is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression 𝑛𝑥 - 2. Therefore, the total cost, in dollars, for 𝑥 hours of repair is 𝑓𝑥 = 220 + 𝑛𝑥 - 2. It’s given that the total cost for 5 hours of repair is $ 400. Substituting 5 for 𝑥 and 400 for 𝑓𝑥 into the equation 𝑓𝑥 = 220 + 𝑛𝑥 - 2 yields 400 = 220 + 𝑛5 - 2, or 400 = 220 + 3𝑛. Subtracting 220 from both sides of this equation yields 180 = 3𝑛. Dividing both sides of this equation by 3 yields 𝑛 = 60. Substituting 60 for 𝑛 in the equation 𝑓𝑥 = 220 + 𝑛𝑥 - 2 yields 𝑓𝑥 = 220 + 60𝑥 - 2, which is equivalent to 𝑓𝑥 = 220 + 60𝑥 - 120, or 𝑓𝑥 = 60𝑥 + 100. Therefore, the total cost, in dollars, for 𝑥 hours of repair is 𝑓𝑥 = 60𝑥 + 100. Choice B is incorrect. This function represents the total cost, in dollars, for 𝑥 hours of repair where the specialist charges $ 340, rather than $ 220, for the first two hours of repair. Choice C is incorrect. This function represents the total cost, in dollars, for 𝑥 hours of repair where the specialist charges $ 160, rather than $ 220, for the first two hours of repair, and an hourly fee of $ 80, rather than $ 60, after the first two hours. Choice D is incorrect. This function represents the total cost, in dollars, for 𝑥 hours of repair where the specialist charges $ 380, rather than $ 220, for the first two hours of repair, and an hourly fee of $ 80, rather than $ 60, after the first two hours. Question Difficulty: Hard Question ID 2937ef4f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: 2937ef4f Hector used a tool called an auger to remove corn from a storage bin at a constant rate. The bin contained 24,000 bushels of corn when Hector began to use the auger. After 5 hours of using the auger, 19,350 bushels of corn remained in the bin. If the auger continues to remove corn at this rate, what is the total number of hours Hector will have been using the auger when 12,840 bushels of corn remain in the bin? A. 3 B. 7 C. 8 D. 12 ID: 2937ef4f Answer Correct Answer: D Rationale Choice D is correct. After using the auger for 5 hours, Hector had removed 24,000 – 19,350 = 4,650 bushels of corn from the storage bin. During the 5-hour period, the auger removed corn from the bin at a constant rate of bushels per hour. Assuming the auger continues to remove corn at this rate, after x hours it will have removed 930x bushels of corn. Because the bin contained 24,000 bushels of corn when Hector started using the auger, the equation 24,000 – 930x = 12,840 can be used to find the number of hours, x, Hector will have been using the auger when 12,840 bushels of corn remain in the bin. Subtracting 12,840 from both sides of this equation and adding 930x to both sides of the equation yields 11,160 = 930x. Dividing both sides of this equation by 930 yields x = 12. Therefore, Hector will have been using the auger for 12 hours when 12,840 bushels of corn remain in the storage bin. Choice A is incorrect. Three hours after Hector began using the auger, 24,000 – 3(930) = 21,210 bushels of corn remained, not 12,840. Choice B is incorrect. Seven hours after Hector began using the auger, 24,000 – 7(930) = 17,490 bushels of corn will remain, not 12,840. Choice C is incorrect. Eight hours after Hector began using the auger, 24,000 – 8(930) = 16,560 bushels of corn will remain, not 12,840. Question Difficulty: Hard Question ID 9bbce683 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 9bbce683 For line , the table shows three values of and their corresponding values of. Line is the result of translating line down units in the xy-plane. What is the x-intercept of line ? A. B. C. D. ID: 9bbce683 Answer Correct Answer: D Rationale Choice D is correct. The equation of line ℎ can be written in slope-intercept form 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 is the slope of the line and 0, 𝑏 is the y-intercept of the line. It’s given that line ℎ contains the points 18, 130, 23, 160, and 26, 178. Therefore, its slope 𝑚 160 - 130 can be found as , or 6. Substituting 6 for 𝑚 in the equation 𝑦= 𝑚𝑥 + 𝑏 yields 𝑦 = 6𝑥 + 𝑏. Substituting 130 for 𝑦 and 18 23 - 18 for 𝑥 in this equation yields 130 = 6 ( 18 ) + 𝑏, or 130 = 108 + 𝑏. Subtracting 108 from both sides of this equation yields 22 = 𝑏. Substituting 22 for 𝑏 in 𝑦 = 6𝑥 + 𝑏 yields 𝑦 = 6𝑥 + 22. Since line 𝑘 is the result of translating line ℎ down 5 units, an equation of line 𝑘 is 𝑦 = 6𝑥 + 22 - 5, or 𝑦 = 6𝑥 + 17. Substituting 0 for 𝑦 in this equation yields 0 = 6𝑥 + 17. Solving this equation for 𝑥 17 17 yields 𝑥 = -. Therefore, the x-intercept of line 𝑘 is - , 0. 6 6 Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID 2b15d65f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: 2b15d65f An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit? A. 16,250 B. 16,500 C. 16,750 D. 17,500 ID: 2b15d65f Answer Correct Answer: A Rationale Choice A is correct. Let the economist’s model be the linear function , where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where. Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs , two points that satisfy the function are and. The slope m of the function can be found using the formula. Substituting the given values into this formula yields , or. Therefore,. The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point yields , or. Adding 10,000 to both sides of this equation yields. Therefore, the linear function the economist used as the model is. Substituting 55 for P yields. It follows that when the selling price is $55 per unit, the demand is 16,250 units. Choices B, C, and D are incorrect and may result from calculation or conceptual errors. Question Difficulty: Hard Question ID 686b7244 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 686b7244 A certain apprentice has enrolled in hours of training courses. The equation represents this situation, where is the number of on-site training courses and is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course? ID: 686b7244 Answer Correct Answer: 5 Rationale The correct answer is 5. It's given that the equation 10𝑥 + 15𝑦 = 85 represents the situation, where 𝑥 is the number of on-site training courses, 𝑦 is the number of online training courses, and 85 is the total number of hours of training courses the apprentice has enrolled in. Therefore, 10𝑥 represents the number of hours the apprentice has enrolled in on-site training courses, and 15𝑦 represents the number of hours the apprentice has enrolled in online training courses. Since 𝑥 is the number of on-site training courses and 𝑦 is the number of online training courses the apprentice has enrolled in, 10 is the number of hours each on-site course takes and 15 is the number of hours each online course takes. Subtracting these numbers gives 15 - 10, or 5 more hours each online training course takes than each on-site training course. Question Difficulty: Hard Question ID be9cb6a2 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: be9cb6a2 The cost of renting a backhoe for up to days is for the first day and for each additional day. Which of the following equations gives the cost , in dollars, of renting the backhoe for days, where is a positive integer and ? A. B. C. D. ID: be9cb6a2 Answer Correct Answer: D Rationale Choice D is correct. It's given that the cost of renting a backhoe for up to 10 days is $ 270 for the first day and $ 135 for each additional day. Therefore, the cost 𝑦, in dollars, for 𝑥 days, where 𝑥 ≤ 10, is the sum of the cost for the first day, $ 270, and the cost for the additional 𝑥 - 1 days, $ 135𝑥 - 1. It follows that 𝑦 = 270 + 135𝑥 - 1, which is equivalent to 𝑦 = 270 + 135𝑥 - 135, or 𝑦 = 135𝑥 + 135. Choice A is incorrect. This equation represents a situation where the cost of renting a backhoe is $ 135 for the first day and $ 270 for each additional day. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID db422e7f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: db422e7f Line is defined by. Line is perpendicular to line in the xy-plane. What is the slope of line ? ID: db422e7f Answer Correct Answer:.5, 1/2 Rationale 1 The correct answer is. For an equation in slope-intercept form 𝑦 = 𝑚𝑥 + 𝑏, 𝑚 represents the slope of the line in the xy-plane 2 defined by this equation. It's given that line 𝑝 is defined by 4𝑦 + 8𝑥 = 6. Subtracting 8𝑥 from both sides of this equation yields 4𝑦 = - 8𝑥 + 6. Dividing both sides of this equation by 4 yields 𝑦 = - 8 𝑥 + 6 , or 𝑦 = - 2𝑥 + 3. Thus, the slope of line 𝑝 is -2. If line 4 4 2 𝑟 is perpendicular to line 𝑝, then the slope of line 𝑟 is the negative reciprocal of the slope of line 𝑝. The negative reciprocal of -2 is - -21 = 12. Note that 1/2 and.5 are examples of ways to enter a correct answer. Question Difficulty: Hard Question ID 45cfb9de Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 45cfb9de Adam’s school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, w, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of w for which it would be faster for Adam to walk to school? A. B. C. D. ID: 45cfb9de Answer Correct Answer: D Rationale Choice D is correct. It is given that w is the number of minutes that Adam waits for the bus. The total time it takes Adam to get to school on a day he takes the bus is the sum of the minutes, w, he waits for the bus and the 5 minutes the bus ride takes; thus, this time, in minutes, is w + 5. It is also given that the total amount of time it takes Adam to get to school on a day that he walks is 20 minutes. Therefore, w + 5 > 20 gives the values of w for which it would be faster for Adam to walk to school. Choices A and B are incorrect because w – 5 is not the total length of time for Adam to wait for and then take the bus to school. Choice C is incorrect because the inequality should be true when walking 20 minutes is faster than the time it takes Adam to wait for and ride the bus, not less. Question Difficulty: Hard Question ID b7e6394d Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: b7e6394d Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week? A. B. C. D. ID: b7e6394d Answer Correct Answer: D Rationale Choice D is correct. Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression gives the cost, in dollars per mile, to drive the car. Multiplying by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his weekly spending by $5, so setting m equal to 5 gives the number of miles, m, by which he must reduce his driving. Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction would result in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to reduce his weekly expenditure by $5. Question Difficulty: Hard Question ID 95cad55f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 95cad55f A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds, and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following systems of inequalities best represents this situation? A. B. C. D. ID: 95cad55f Answer Correct Answer: A Rationale Choice A is correct. The number of containers in a shipment must have a weight less than or equal to 300 pounds. The total weight, in pounds, of detergent and fabric softener that the supplier delivers can be expressed as the weight of each container multiplied by the number of each type of container, which is 7.35d for detergent and 6.2s for fabric softener. Since this total cannot exceed 300 pounds, it follows that. Also, since the laundry service wants to buy at least twice as many containers of detergent as containers of fabric softener, the number of containers of detergent should be greater than or equal to two times the number of containers of fabric softener. This can be expressed by the inequality. Choice B is incorrect because it misrepresents the relationship between the numbers of each container that the laundry service wants to buy. Choice C is incorrect because the first inequality of the system incorrectly doubles the weight per container of detergent. The weight of each container of detergent is 7.35, not 14.7 pounds. Choice D is incorrect because it doubles the weight per container of detergent and transposes the relationship between the numbers of containers. Question Difficulty: Hard Question ID ee2f611f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: ee2f611f A local transit company sells a monthly pass for $95 that allows an unlimited number of trips of any length. Tickets for individual trips cost $1.50, $2.50, or $3.50, depending on the length of the trip. What is the minimum number of trips per month for which a monthly pass could cost less than purchasing individual tickets for trips? ID: ee2f611f Answer Rationale The correct answer is 28. The minimum number of individual trips for which the cost of the monthly pass is less than the cost of individual tickets can be found by assuming the maximum cost of the individual tickets, $3.50. If n tickets costing $3.50 each are purchased in one month, the inequality 95 < 3.50n represents this situation. Dividing both sides of the inequality by 3.50 yields 27.14 < n, which is equivalent to n > 27.14. Since only a whole number of tickets can be purchased, it follows that 28 is the minimum number of trips. Question Difficulty: Hard Question ID 25e1cfed Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: 25e1cfed How many solutions does the equation have? A. Exactly one B. Exactly two C. Infinitely many D. Zero ID: 25e1cfed Answer Correct Answer: C Rationale Choice C is correct. Applying the distributive property to each side of the given equation yields 150𝑥 - 90 = - 90 + 150𝑥. Applying the commutative property of addition to the right-hand side of this equation yields 150𝑥 - 90 = 150𝑥 - 90. Since the two sides of the equation are equivalent, this equation is true for any value of 𝑥. Therefore, the given equation has infinitely many solutions. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID fdee0fbf Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: fdee0fbf In the xy-plane, line k intersects the y-axis at the point and passes through the point. If the point lies on line k, what is the value of w ? ID: fdee0fbf Answer Rationale The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair of the point of intersection of the line with the y-axis. Since line k intersects the y-axis at the point , it follows that is the y-intercept of this line. An equation of any line in the xy-plane can be written in the form , where m is the slope of the line and b is the y- coordinate of the y-intercept. Therefore, the equation of line k can be written as , or. The value of m can be found by substituting the x- and y-coordinates from a point on the line, such as , for x and y, respectively. This results in. Solving this equation for m gives. Therefore, an equation of line k is. The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for y. This gives , or. Since w is the y-coordinate of this point,. Question Difficulty: Hard Question ID 541bef2f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 541bef2f Which point is a solution to the given system of inequalities in the xy-plane? A. B. C. D. ID: 541bef2f Answer Correct Answer: D Rationale Choice D is correct. A point 𝑥, 𝑦 is a solution to a system of inequalities in the xy-plane if substituting the x-coordinate and the y-coordinate of the point for 𝑥 and 𝑦, respectively, in each inequality makes both of the inequalities true. Substituting the x-coordinate and the y-coordinate of choice D, 14 and 0, for 𝑥 and 𝑦, respectively, in the first inequality in the given system, 𝑦 ≤ 𝑥 + 7, yields 0 ≤ 14 + 7, or 0 ≤ 21, which is true. Substituting 14 for 𝑥 and 0 for 𝑦 in the second inequality in the given system, 𝑦 ≥ - 2𝑥 - 1, yields 0 ≥ - 214 - 1, or 0 ≥ - 29, which is true. Therefore, the point 14, 0 is a solution to the given system of inequalities in the xy-plane. Choice A is incorrect. Substituting -14 for 𝑥 and 0 for 𝑦 in the inequality 𝑦 ≤ 𝑥 + 7 yields 0 ≤ - 14 + 7, or 0 ≤ - 7, which is not true. Choice B is incorrect. Substituting 0 for 𝑥 and -14 for 𝑦 in the inequality 𝑦 ≥ - 2𝑥 - 1 yields -14 ≥ - 20 - 1, or -14 ≥ - 1, which is not true. Choice C is incorrect. Substituting 0 for 𝑥 and 14 for 𝑦 in the inequality 𝑦 ≤ 𝑥 + 7 yields 14 ≤ 0 + 7, or 14 ≤ 7, which is not true. Question Difficulty: Hard Question ID f75bd744 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: f75bd744 In the given system of equations, is a constant. If the system has no solution, what is the value of ? ID: f75bd744 Answer Correct Answer: 8 Rationale The correct answer is 8. The given system of equations can be solved using the elimination method. Multiplying both sides of the second equation in the given system by -2 yields -2𝑡𝑦 = - 1 - 4𝑥, or -1 - 4𝑥 = - 2𝑡𝑦. Adding this equation to the first equation in the given system, 4𝑥 - 6𝑦 = 10𝑦 + 2, yields 4𝑥 - 6𝑦 + -1 - 4𝑥 = 10𝑦 + 2 + -2𝑡𝑦, or -1 - 6𝑦 = 10𝑦 - 2𝑡𝑦 + 2. Subtracting 10𝑦 from both sides of this equation yields -1 - 6𝑦 - 10𝑦 = 10𝑦 - 2𝑡𝑦 + 2 - 10𝑦, or -1 - 16𝑦 = - 2𝑡𝑦 + 2. If the given system has no solution, then the equation -1 - 16𝑦 = - 2𝑡𝑦 + 2 has no solution. If this equation has no solution, the coefficients of 𝑦 on each side of the equation, -16 and -2𝑡, must be equal, which yields the equation -16 = - 2𝑡. Dividing both sides of this equation by -2 yields 8 = 𝑡. Thus, if the system has no solution, the value of 𝑡 is 8. Alternate approach: A system of two linear equations in two variables, 𝑥 and 𝑦, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in the form 𝐴𝑥 + 𝐵𝑦 = 𝐶, where 𝐴, 𝐵, and 𝐶 are constant terms, are parallel if the ratio of the x-coefficients is equal to the ratio of the y-coefficients, and distinct if the ratio of the x-coefficients are not equal to the ratio of the constant terms. Subtracting 10𝑦 from both sides of the first equation in the given system yields 4𝑥 - 6𝑦 - 10𝑦 = 10𝑦 + 2 - 10𝑦, or 4𝑥 - 16𝑦 = 2. Subtracting 2𝑥 from both sides of the second 1 1 equation in the given system yields 𝑡𝑦 - 2𝑥 = + 2𝑥 - 2𝑥, or -2𝑥 + 𝑡𝑦 =. The ratio of the x-coefficients for these equations is 2 2 1 - 4 , or - 2. The ratio of the y-coefficients for these equations is - 16. The ratio of the constant terms for these equations is 22 , or 14 2 1 𝑡 1 1. Since the ratio of the x-coefficients, - , is not equal to the ratio of the constants, , the lines represented by the equations are 2 4 1 𝑡 distinct. Setting the ratio of the x-coefficients equal to the ratio of the y-coefficients yields - = -. Multiplying both sides of 2 16 1 𝑡 this equation by -16 yields - -16 = - -16, or 𝑡 = 8. Therefore, when 𝑡 = 8, the lines represented by these equations are parallel. 2 16 Thus, if the system has no solution, the value of 𝑡 is 8. Question Difficulty: Hard Question ID b3abf40f Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: b3abf40f The function gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of kelvins. If a temperature increased by kelvins, by how much did the temperature increase, in degrees Fahrenheit? A. B. C. D. ID: b3abf40f Answer Correct Answer: A Rationale 9 Choice A is correct. It’s given that the function 𝐹𝑥 = 𝑥 - 273.15 + 32 gives the temperature, in degrees Fahrenheit, that 5 corresponds to a temperature of 𝑥 kelvins. A temperature that increased by 9.10 kelvins means that the value of 𝑥 increased 9 by 9.10 kelvins. It follows that an increase in 𝑥 by 9.10 increases 𝐹 ( 𝑥 ) by 9.10, or 16.38. Therefore, if a temperature increased 5 by 9.10 kelvins, the temperature increased by 16.38 degrees Fahrenheit. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID e6cb2402 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: e6cb2402 In the given equation, is a constant. The equation has no solution. What is the value of ? ID: e6cb2402 Answer Correct Answer:.9411,.9412, 16/17 Rationale 16 The correct answer is 17. It's given that the equation 3𝑘𝑥 + 13 = 48 17 𝑥 + 36 has no solution. A linear equation in the form 𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑, where 𝑎, 𝑏, 𝑐, and 𝑑 are constants, has no solution only when the coefficients of 𝑥 on each side of the equation are equal and the constant terms aren't equal. Dividing both sides of the given equation by 3 yields 𝑘𝑥 + 13 = 48 51 𝑥 + 363 , or 𝑘𝑥 + 13 = 16 17 𝑥 + 12. Since the coefficients of 𝑥 on each side of the equation must be equal, it follows 16 that the value of 𝑘 is. Note that 16/17,.9411,.9412, and 0.941 are examples of ways to enter a correct answer. 17 Question Difficulty: Hard Question ID 6c71f3ec Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 6c71f3ec A salesperson’s total earnings consist of a base salary of dollars per year, plus commission earnings of of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least times and at most times the base salary. Which of the following inequalities represents all possible values of total sales , in dollars, the salesperson can make this year in order to meet that goal? A. B. C. D. ID: 6c71f3ec Answer Correct Answer: B Rationale Choice B is correct. It’s given that a salesperson's total earnings consist of a base salary of 𝑥 dollars per year plus commission earnings of 11% of the total sales the salesperson makes during the year. If the salesperson makes 𝑠 dollars in total sales this year, the salesperson’s total earnings can be represented by the expression 𝑥 + 0.11𝑠. It’s also given that the salesperson has a goal for the total earnings to be at least 3 times and at most 4 times the base salary, which can be represented by the expressions 3𝑥 and 4𝑥, respectively. Therefore, this situation can be represented by the inequality 3𝑥 ≤ 𝑥 + 0.11𝑠 ≤ 4𝑥. Subtracting 𝑥 from each part of this inequality yields 2𝑥 ≤ 0.11𝑠 ≤ 3𝑥. Dividing each part of this 2 inequality by 0.11 yields 𝑥 ≤ 𝑠 ≤ 3 𝑥. Therefore, the inequality 2 𝑥 ≤ 𝑠 ≤ 3 𝑥 represents all possible values of total sales 0.11 0.11 0.11 0.11 𝑠, in dollars, the salesperson can make this year in order to meet their goal. Choice A is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least 2 times and at most 3 times, rather than at least 3 times and at most 4 times, the base salary. Choice C is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least 3 times and at most 4 times the base salary. Choice D is incorrect. This inequality represents a situation in which the total earnings are at least 4 times and at most 5 times, rather than at least 3 times and at most 4 times, the base salary. Question Difficulty: Hard Question ID b988eeec Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: b988eeec The functions and are defined as and. If the function is defined as , what is the x-coordinate of the x-intercept of the graph of in the xy-plane? ID: b988eeec Answer Correct Answer: -12 Rationale 1 The correct answer is -12. It's given that the functions 𝑓 and 𝑔 are defined as 𝑓𝑥 = 4 = 34 𝑥 + 21. If the function ℎ is 𝑥 - 9 and 𝑔𝑥 1 3 1 3 defined as ℎ𝑥 = 𝑓𝑥 + 𝑔𝑥, then substituting 𝑥 - 9 for 𝑓𝑥 and 𝑥 + 21 for 𝑔𝑥 in this function yields ℎ𝑥 = 𝑥 - 9 + 𝑥 + 21. This 4 4 4 4 4 can be rewritten as ℎ𝑥 = 𝑥 + 12, or ℎ𝑥 = 𝑥 + 12. The x-intercept of a graph in the xy-plane is the point on the graph where 4 𝑦 = 0. The equation representing the graph of 𝑦 = ℎ𝑥 is 𝑦 = 𝑥 + 12. Substituting 0 for 𝑦 in this equation yields 0 = 𝑥 + 12. Subtracting 12 from both sides of this equation yields -12 = 𝑥, or 𝑥 = - 12. Therefore, the x-coordinate of the x-intercept of the graph of 𝑦 = ℎ𝑥 in the xy-plane is -12. Question Difficulty: Hard Question ID 70feb725 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: 70feb725 During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per hour. She ran and biked a total of 200 miles that month, and she biked for twice as many hours as she ran. What is the total number of miles that Morgan biked during the month? A. 80 B. 100 C. 120 D. 160 ID: 70feb725 Answer Correct Answer: D Rationale Choice D is correct. The number of hours Morgan spent running or biking can be calculated by dividing the distance she traveled during that activity by her speed, in miles per hour, for that activity. So the number of hours she ran can be represented by the expression , and the number of hours she biked can be represented by the expression. It’s given that she biked for twice as many hours as she ran, so this can be represented by the equation , which can be rewritten as. It’s also given that she ran r miles and biked b miles, and that she ran and biked a total of 200 miles. This can be represented by the equation. Substituting for b in this equation yields , or. Solving for r yields. Determining the number of miles she biked, b, can be found by substituting 40 for r in , which yields. Solving for b yields. Choices A, B, and C are incorrect because they don’t satisfy that Morgan biked for twice as many hours as she ran. In choice A, if she biked 80 miles, then she ran 120 miles, which means she biked for 8 hours and ran for 24 hours. In choice B, if she biked 100 miles, then she ran 100 miles, which means she biked for 10 hours and ran for 20 hours. In choice C, if she biked 120 miles, then she ran for 80 miles, which means she biked for 12 hours and ran for 16 hours. Question Difficulty: Hard Question ID 1a621af4 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 1a621af4 A number is at most less than times the value of. If the value of is , what is the greatest possible value of ? ID: 1a621af4 Answer Correct Answer: -14 Rationale The correct answer is -14. It's given that a number 𝑥 is at most 2 less than 3 times the value of 𝑦. Therefore, 𝑥 is less than or equal to 2 less than 3 times the value of 𝑦. The expression 3𝑦 represents 3 times the value of 𝑦. The expression 3𝑦 - 2 represents 2 less than 3 times the value of 𝑦. Therefore, 𝑥 is less than or equal to 3𝑦 - 2. This can be shown by the inequality 𝑥 ≤ 3𝑦 - 2. Substituting -4 for 𝑦 in this inequality yields 𝑥 ≤ 3-4 - 2 or, 𝑥 ≤ - 14. Therefore, if the value of 𝑦 is -4, the greatest possible value of 𝑥 is -14. Question Difficulty: Hard Question ID af2ba762 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: af2ba762 According to data provided by the US Department of Energy, the average price per gallon of regular gasoline in the United States from September 1, 2014, to December 1, 2014, is modeled by the function F defined below, where is the average price per gallon x months after September 1. The constant 2.74 in this function estimates which of the following? A. The average monthly decrease in the price per gallon B. The difference in the average price per gallon from September 1, 2014, to December 1, 2014 C. The average price per gallon on September 1, 2014 D. The average price per gallon on December 1, 2014 ID: af2ba762 Answer Correct Answer: D Rationale Choice D is correct. Since 2.74 is a constant term, it represents an actual price of gas rather than a measure of change in gas price. To determine what gas price it represents, find x such that F(x) = 2.74, or 2.74 = 2.74 – 0.19(x – 3). Subtracting 2.74 from both sides gives 0 = –0.19(x – 3). Dividing both sides by –0.19 results in 0 = x – 3, or x = 3. Therefore, the average price of gas is $2.74 per gallon 3 months after September 1, 2014, which is December 1, 2014. Choice A is incorrect. Since 2.74 is a constant, not a multiple of x, it cannot represent a rate of change in price. Choice B is incorrect. The difference in the average price from September 1, 2014, to December 1, 2014, is F(3) – F(0) = 2.74 – 0.19(3 – 3) – (2.74 – 0.19(0 – 3)) = 2.74 – (2.74 + 0.57) = –0.57, which is not 2.74. Choice C is incorrect. The average price per gallon on September 1, 2014, is F(0) = 2.74 – 0.19(0 – 3) = 2.74 + 0.57 = 3.31, which is not 2.74. Question Difficulty: Hard Question ID b9835972 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: b9835972 In the xy-plane, line 𝓁 passes through the point and is parallel to the line represented by the equation. If line 𝓁 also passes through the point , what is the value of ? ID: b9835972 Answer Correct Answer: 24 Rationale The correct answer is 24. A line in the xy-plane can be defined by the equation 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 is the slope of the line and 𝑏 is the y-coordinate of the y-intercept of the line. It's given that line 𝑙 passes through the point 0, 0. Therefore, the y- coordinate of the y-intercept of line 𝑙 is 0. It's given that line 𝑙 is parallel to the line represented by the equation 𝑦 = 8𝑥 + 2. Since parallel lines have the same slope, it follows that the slope of line 𝑙 is 8. Therefore, line 𝑙 can be defined by an equation in the form 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 = 8 and 𝑏 = 0. Substituting 8 for 𝑚 and 0 for 𝑏 in 𝑦 = 𝑚𝑥 + 𝑏 yields the equation 𝑦 = 8𝑥 + 0, or 𝑦 = 8𝑥. If line 𝑙 passes through the point 3, 𝑑, then when 𝑥 = 3, 𝑦 = 𝑑 for the equation 𝑦 = 8𝑥. Substituting 3 for 𝑥 and 𝑑 for 𝑦 in the equation 𝑦 = 8𝑥 yields 𝑑 = 83, or 𝑑 = 24. Question Difficulty: Hard Question ID e1248a5c Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: e1248a5c In the system of equations below, a and c are constants. If the system of equations has an infinite number of solutions , what is the value of a ? A. B. 0 C. D. ID: e1248a5c Answer Correct Answer: D Rationale Choice D is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side and a constant on the right-hand side of the equation. The coefficient of y in the second equation is equal to the coefficient of y in the first equation multiplied by 3. Therefore, a, the coefficient of x in the second equation, must be equal to 3 times the coefficient of x in the first equation: , or. Choices A, B, and C are incorrect. When , , or , the given system of equations has one solution. Question Difficulty: Hard Question ID 05bb1af9 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 05bb1af9 The graph of is shown. Which equation defines function ? A. B. C. D. ID: 05bb1af9 Answer Correct Answer: A Rationale Choice A is correct. An equation for the graph shown can be written in slope-intercept form 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 is the slope of the graph and its y-intercept is 0, 𝑏. Since the y-intercept of the graph shown is 0, 2, the value of 𝑏 is 2. Since the graph also 1-2 1 1 1 passes through the point 4, 1, the slope can be calculated as , or -. Therefore, the value of 𝑚 is -. Substituting - for 𝑚 4-0 4 4 4 and 2 for 𝑏 in the equation 𝑦 = 𝑚𝑥 + 𝑏 yields 𝑦 = - 14 𝑥 + 2. It’s given that an equation for the graph shown is 𝑦 = 𝑓𝑥 + 14. 1 1 Substituting 𝑓𝑥 + 14 for 𝑦 in the equation 𝑦 = - 𝑥 + 2 yields 𝑓𝑥 + 14 = - 𝑥 + 2. Subtracting 14 from both sides of this 4 4 1 equation yields 𝑓𝑥 = - 𝑥 - 12. 4 Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID cc7ffe02 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: cc7ffe02 Keenan made cups of vegetable broth. Keenan then filled small jars and large jars with all the vegetable broth he made. The equation represents this situation. Which is the best interpretation of in this context? A. The number of large jars Keenan filled B. The number of small jars Keenan filled C. The total number of cups of vegetable broth in the large jars D. The total number of cups of vegetable broth in the small jars ID: cc7ffe02 Answer Correct Answer: C Rationale Choice C is correct. It’s given that the equation 3𝑥 + 5𝑦 = 32 represents the situation where Keenan filled 𝑥 small jars and 𝑦 large jars with all the vegetable broth he made, which was 32 cups. Therefore, 3𝑥 represents the total number of cups of vegetable broth in the small jars and 5𝑦 represents the total number of cups of vegetable broth in the large jars. Choice A is incorrect. The number of large jars Keenan filled is represented by 𝑦, not 5𝑦. Choice B is incorrect. The number of small jars Keenan filled is represented by 𝑥, not 5𝑦. Choice D is incorrect. The total number of cups of vegetable broth in the small jars is represented by 3𝑥, not 5𝑦. Question Difficulty: Hard Question ID ae2287e2 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: ae2287e2 A certain product costs a company $65 to make. The product is sold by a salesperson who earns a commission that is equal to 20% of the sales price of the product. The profit the company makes for each unit is equal to the sales price minus the combined cost of making the product and the commission. If the sales price of the product is $100, which of the following equations gives the number of units, u, of the product the company sold to make a profit of $6,840 ? A. B. C. D. ID: ae2287e2 Answer Correct Answer: A Rationale Choice A is correct. The sales price of one unit of the product is given as $100. Because the salesperson is awarded a commission equal to 20% of the sales price, the expression 100(1 – 0.2) gives the sales price of one unit after the commission is deducted. It is also given that the profit is equal to the sales price minus the combined cost of making the product, or $65, and the commission: 100(1 – 0.2) – 65. Multiplying this expression by u gives the profit of u units: (100(1 – 0.2) – 65)u. Finally, it is given that the profit for u units is $6,840; therefore (100(1 – 0.2) – 65)u = $6,840. Choice B is incorrect. In this equation, cost is subtracted before commission and the equation gives the commission, not what the company retains after commission. Choice C is incorrect because the number of units is multiplied only by the cost but not by the sale price. Choice D is incorrect because the value 0.2 shows the commission, not what the company retains after commission. Question Difficulty: Hard Question ID 1362ccde Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: 1362ccde The solution to the given system of equations is. What is the value of ? ID: 1362ccde Answer Correct Answer: 35 Rationale The correct answer is 35. The first equation in the given system of equations defines 𝑦 as 4𝑥 + 1. Substituting 4𝑥 + 1 for 𝑦 in the second equation in the given system of equations yields 44𝑥 + 1 = 15𝑥 - 8. Applying the distributive property on the left- hand side of this equation yields 16𝑥 + 4 = 15𝑥 - 8. Subtracting 15𝑥 from each side of this equation yields 𝑥 + 4 = - 8. Subtracting 4 from each side of this equation yields 𝑥 = - 12. Substituting -12 for 𝑥 in the first equation of the given system of equations yields 𝑦 = 4-12 + 1, or 𝑦 = - 47. Substituting -12 for 𝑥 and -47 for 𝑦 into the expression 𝑥 - 𝑦 yields -12 - -47, or 35. Question Difficulty: Hard Question ID 52cb8ea4 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: 52cb8ea4 If is the solution to the system of equations above, what is the value of ? A. B. C. 5 D. 13 ID: 52cb8ea4 Answer Correct Answer: B Rationale Choice B is correct. Subtracting the second equation, , from the first equation, , results in , or. Combining like terms on the left-hand side of this equation yields. Choice A is incorrect and may result from miscalculating as. Choice C is incorrect and may result from miscalculating as 5. Choice D is incorrect and may result from adding 9 to 4 instead of subtracting 9 from 4. Question Difficulty: Hard Question ID 0b46bad5 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 0b46bad5 In the equation above, a and b are constants and. Which of the following could represent the graph of the equation in the xy-plane? A. B. C. D. ID: 0b46bad5 Answer Correct Answer: C Rationale Choice C is correct. The given equation can be rewritten in slope-intercept form, , where m represents the slope of the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the equation yields , and dividing both sides of this equation by b yields , or. With the equation now in slope-intercept form, it shows that , which means the y- coordinate of the y-intercept is 1. It’s given that a and b are both greater than 0 (positive) and that. Since , the slope of the line must be a value between and 0. Choice C is the only graph of a line that has a y-value of the y- intercept that is 1 and a slope that is between and 0. Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between and 0. Question Difficulty: Hard Question ID 7d5d1b32 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: 7d5d1b32 In the given equation, and are constants and. The equation has no solution. What is the value of ? ID: 7d5d1b32 Answer Correct Answer: -.9333, -14/15 Rationale 14 The correct answer is -. A linear equation in the form 𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑 has no solution only when the coefficients of 𝑥 on 15 each side of the equation are equal and the constant terms are not equal. Dividing both sides of the given equation by 2 28 36 14 18 yields 𝑘𝑥 - 𝑛 = - 𝑥 - , or 𝑘𝑥 - 𝑛 = - 𝑥 -. Since it’s given that the equation has no solution, the coefficient of 𝑥 on both 30 38 15 19 18 sides of this equation must be equal, and the constant terms on both sides of this equation must not be equal. Since < 1, 19 and it's given that 𝑛 > 1, the second condition is true. Thus, 𝑘 must be equal to - 14 15. Note that -14/15, -.9333, and -0.933 are examples of ways to enter a correct answer. Question Difficulty: Hard Question ID c362c210 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: c362c210 The points plotted in the coordinate plane above represent the possible numbers of wallflowers and cornflowers that someone can buy at the Garden Store in order to spend exactly $24.00 total on the two types of flowers. The price of each wallflower is the same and the price of each cornflower is the same. What is the price, in dollars, of 1 cornflower? ID: c362c210 Answer Rationale The correct answer is 1.5. The point corresponds to the situation where 16 cornflowers and 0 wallflowers are purchased. Since the total spent on the two types of flowers is $24.00, it follows that the price of 16 cornflowers is $24.00, and the price of one cornflower is $1.50. Note that 1.5 and 3/2 are examples of ways to enter a correct answer. Question Difficulty: Hard Question ID ee7b1de1 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: ee7b1de1 A small business owner budgets to purchase candles. The owner must purchase a minimum of candles to maintain the discounted pricing. If the owner pays per candle to purchase small candles and per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing? ID: ee7b1de1 Answer Correct Answer: 182 Rationale The correct answer is 182. Let 𝑠 represent the number of small candles the owner can purchase, and let 𝑙 represent the number of large candles the owner can purchase. It’s given that the owner pays $ 4.90 per candle to purchase small candles and $ 11.60 per candle to purchase large candles. Therefore, the owner pays 4.90𝑠 dollars for 𝑠 small candles and 11.60𝑙 dollars for 𝑙 large candles, which means the owner pays a total of 4.90𝑠 + 11.60𝑙 dollars to purchase candles. It’s given that the owner budgets $ 2,200 to purchase candles. Therefore, 4.90𝑠 + 11.60𝑙 ≤ 2,200. It’s also given that the owner must purchase a minimum of 200 candles. Therefore, 𝑠 + 𝑙 ≥ 200. The inequalities 4.90𝑠 + 11.60𝑙 ≤ 2,200 and 𝑠 + 𝑙 ≥ 200 can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left- hand side of the first inequality. Subtracting 𝑙 from both sides of the inequality 𝑠 + 𝑙 ≥ 200 yields 𝑠 ≥ 200 - 𝑙. Multiplying both sides of this inequality by 4.90 yields 4.90𝑠 ≥ 4.90200 - 𝑙, or 4.90𝑠 ≥ 980 - 4.90𝑙. Adding 11.60𝑙 to both sides of this inequality yields 4.90𝑠 + 11.60𝑙 ≥ 980 - 4. 90𝑙 + 11. 60𝑙, or 4.90𝑠 + 11.60𝑙 ≥ 980 + 6.70𝑙. This inequality can be combined with the inequality 4.90𝑠 + 11.60𝑙 ≤ 2,200, which yields the compound inequality 980 + 6.70𝑙 ≤ 4.90𝑠 + 11.60𝑙 ≤ 2,200. It follows that 980 + 6.70𝑙 ≤ 2,200. Subtracting 980 from both sides of this inequality yields 6.70𝑙 ≤ 2,200. Dividing both sides of this inequality by 6.70 yields approximately 𝑙 ≤ 182.09. Since the number of large candles the owner purchases must be a whole number, the maximum number of large candles the owner can purchase is the largest whole number less than 182.09, which is 182. Question Difficulty: Hard Question ID 94b48cbf Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 94b48cbf The graph of in the -plane has an -intercept at and a -intercept at , where and are constants. What is the value of ? A. B. C. D. ID: 94b48cbf Answer Correct Answer: D Rationale Choice D is correct. The x-coordinate 𝑎 of the x-intercept 𝑎, 0 can be found by substituting 0 for 𝑦 in the given equation, which gives 7𝑥 + 20 = - 31, or 7𝑥 = - 31. Dividing both sides of this equation by 7 yields 𝑥 = - 317. Therefore, the value of 𝑎 is - 317. The y-coordinate 𝑏 of the y-intercept 0, 𝑏 can be found by substituting 0 for 𝑥 in the given equation, which gives 70 + 2𝑦 = - 31, or 2𝑦 = - 31. Dividing both sides of this equation by 2 yields 𝑦 = - 312. Therefore, the value of 𝑏 is - 312. It follows that the value of 𝑎𝑏 31 - is 312 , which is equivalent to 31 7 , or 7. - 2 31 2 7 Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID 1035faea Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 1035faea A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 chose the first picture in the set. Among the remaining 150 participants, p people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of p ? A. , where B. , where C. , where D. , where ID: 1035faea Answer Correct Answer: D Rationale Choice D is correct. Of the first 150 participants, 36 chose the first picture in the set, and of the 150 remaining participants, p chose the first picture in the set. Hence, the proportion of the participants who chose the first picture in the set is. Since more than 20% of all the participants chose the first picture, it follows that. This inequality can be rewritten as. Since p is a number of people among the remaining 150 participants,. Choices A, B, and C are incorrect and may be the result of some incorrect interpretations of the given information or of computational errors. Question Difficulty: Hard Question ID 45a534d0 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: 45a534d0 In the given system of equations, is a constant. If the system has no solution, what is the value of ? ID: 45a534d0 Answer Correct Answer: -34 Rationale The correct answer is -34. A system of two linear equations in two variables, 𝑥 and 𝑦, has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. Two lines represented by equations in standard form 𝐴𝑥 + 𝐵𝑦 = 𝐶, where 𝐴, 𝐵, and 𝐶 are constants, are parallel if the coefficients for 𝑥 and 𝑦 in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in standard form by subtracting 30𝑦 from both sides of the equation to yield 48𝑥 - 102𝑦 = 24. The second equation in the given system can be 1 written in standard form by adding 16𝑥 to both sides of the equation to yield 16𝑥 + 𝑟𝑦 =. The coefficient of 𝑥 in this second 6 1 equation, 16, is times the coefficient of 𝑥 in the first equation, 48. For the lines to be parallel the coefficient of 𝑦 in the 3 1 1 second equation, 𝑟, must also be times the coefficient of 𝑦 in the first equation, -102. Thus, 𝑟 = ( - 102) , or 𝑟 = - 34. 3 3 Therefore, if the given system has no solution, the value of 𝑟 is -34. Question Difficulty: Hard Question ID 50f4cb9c Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: 50f4cb9c For the linear function , the table shows three values of and their corresponding values of. Function is defined by , where and are constants. What is the value of ? A. B. C. D. ID: 50f4cb9c Answer Correct Answer: D Rationale Choice D is correct. The table gives that 𝑓 ( 𝑥 ) = 0 when 𝑥 = 2. Substituting 0 for 𝑓𝑥 and 2 for 𝑥 into the equation 𝑓 ( 𝑥 ) = 𝑎𝑥 + 𝑏 yields 0 = 2𝑎 + 𝑏. Subtracting 2𝑎 from both sides of this equation yields 𝑏 = - 2𝑎. The table gives that 𝑓 ( 𝑥 ) = - 64 when 𝑥 = 1. Substituting -2𝑎 for 𝑏, -64 for 𝑓𝑥, and 1 for 𝑥 into the equation 𝑓 ( 𝑥 ) = 𝑎𝑥 + 𝑏 yields -64 = 𝑎1 + -2𝑎. Combining like terms yields -64 = - 𝑎, or 𝑎 = 64. Since 𝑏 = - 2𝑎, substituting 64 for 𝑎 into this equation gives 𝑏 = -264, which yields 𝑏 = - 128. Thus, the value of 𝑎 - 𝑏 can be written as 64 - ( - 128 ) , which is 192. Choice A is incorrect. This is the value of 𝑎 + 𝑏, not 𝑎 - 𝑏. Choice B is incorrect. This is the value of 𝑎 - 2, not 𝑎 - 𝑏. Choice C is incorrect. This is the value of 2𝑎, not 𝑎 - 𝑏. Question Difficulty: Hard Question ID 16889ef3 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: 16889ef3 Oil and gas production in a certain area dropped from 4 million barrels in 2000 to 1.9 million barrels in 2013. Assuming that the oil and gas production decreased at a constant rate, which of the following linear functions f best models the production, in millions of barrels, t years after the year 2000? A. B. C. D. ID: 16889ef3 Answer Correct Answer: C Rationale Choice C is correct. It is assumed that the oil and gas production decreased at a constant rate. Therefore, the function f that best models the production t years after the year 2000 can be written as a linear function, , where m is the rate of change of the oil and gas production and b is the oil and gas production, in millions of barrels, in the year 2000. Since there were 4 million barrels of oil and gas produced in 2000,. The rate of change, m, can be calculated as , which is equivalent to , the rate of change in choice C. Choices A and B are incorrect because each of these functions has a positive rate of change. Since the oil and gas production decreased over time, the rate of change must be negative. Choice D is incorrect. This model may result from misinterpreting 1.9 million barrels as the amount by which the production decreased. Question Difficulty: Hard Question ID adb0c96c Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: adb0c96c The solution to the given system of equations is. What is the value of ? ID: adb0c96c Answer Correct Answer: 80 Rationale The correct answer is 80. Subtracting the second equation in the given system from the first equation yields 24𝑥 + 𝑦 - 6𝑥 + 𝑦 = 48 - 72, which is equivalent to 24𝑥 - 6𝑥 + 𝑦 - 𝑦 = - 24, or 18𝑥 = - 24. Dividing each side of this equation by 3 yields 6𝑥 = - 8. Substituting -8 for 6𝑥 in the second equation yields -8 + 𝑦 = 72. Adding 8 to both sides of this equation yields 𝑦 = 80. Alternate approach: Multiplying each side of the second equation in the given system by 4 yields 24𝑥 + 4𝑦 = 288. Subtracting the first equation in the given system from this equation yields 24𝑥 + 4𝑦 - 24𝑥 + 𝑦 = 288 - 48, which is equivalent to 24𝑥 - 24𝑥 + 4𝑦 - 𝑦 = 240, or 3𝑦 = 240. Dividing each side of this equation by 3 yields 𝑦 = 80. Question Difficulty: Hard Question ID d7bf55e1 Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: d7bf55e1 A movie theater sells two types of tickets, adult tickets for $12 and child tickets for $8. If the theater sold 30 tickets for a total of $300, how much, in dollars, was spent on adult tickets? (Disregard the $ sign when gridding your answer.) ID: d7bf55e1 Answer Rationale The correct answer is 180. Let a be the number of adult tickets sold and c be the number of child tickets sold. Since the theater sold a total of 30 tickets, a + c = 30. The price per adult ticket is $12, and the price per child ticket is $8. Since the theater received a total of $300 for the 30 tickets sold, it follows that 12a + 8c = 300. To eliminate c, the first equation can be multiplied by 8 and then subtracted from the second equation: Because the question asks for the amount spent on adult tickets, which is 12a dollars, the resulting equation can be multiplied by 3 to give 3(4a) = 3(60) = 180. Therefore, $180 was spent on adult tickets. Alternate approach: If all the 30 tickets sold were child tickets, their total price would be 30($8) = $240. Since the actual total price of the 30 tickets was $300, the extra $60 indicates that a certain number of adult tickets, a, were sold. Since the price of each adult ticket is $4 more than each child ticket, 4a = 60, and it follows that 12a = 180. Question Difficulty: Hard Question ID 771bd0ca Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: 771bd0ca What value of is the solution to the given equation? ID: 771bd0ca Answer Correct Answer: -22 Rationale The correct answer is -22. The given equation can be rewritten as -2𝑡 + 3 = 38. Dividing both sides of this equation by -2 yields 𝑡 + 3 = - 19. Subtracting 3 from both sides of this equation yields 𝑡 = - 22. Therefore, -22 is the value of 𝑡 that is the solution to the given equation. Question Difficulty: Hard Question ID a309803e Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: a309803e One gallon of paint will cover square feet of a surface. A room has a total wall area of square feet. Which equation represents the total amount of paint , in gallons, needed to paint the walls of the room twice? A. B. C. D. ID: a309803e Answer Correct Answer: A Rationale Choice A is correct. It's given that 𝑤 represents the total wall area, in square feet. Since the walls of the room will be painted twice, the amount of paint, in gallons, needs to cover 2𝑤 square feet. It’s also given that one gallon of paint will cover 220 square feet. Dividing the total area, in square feet, of the surface to be painted by the number of square feet covered by one 2𝑤 𝑤 gallon of paint gives the number of gallons of paint that will be needed. Dividing 2𝑤 by 220 yields , or. Therefore, the 220 110 𝑤 equation that represents the total amount of paint 𝑃, in gallons, needed to paint the walls of the room twice is 𝑃 =. 110 Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from finding the amount of paint needed to paint the walls once rather than twice. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID 5bf5136d Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 5bf5136d The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of and , which inequality represents the possible lengths, , of the third side of the triangle? A. B. C. D. or ID: 5bf5136d Answer Correct Answer: C Rationale Choice C is correct. It’s given that a triangle has side lengths of 6 and 12, and 𝑥 represents the length of the third side of the triangle. It’s also given that the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. Therefore, the inequalities 6 + 𝑥 > 12, 6 + 12 > 𝑥, and 12 + 𝑥 > 6 represent all possible values of 𝑥. Subtracting 6 from both sides of the inequality 6 + 𝑥 > 12 yields 𝑥 > 12 - 6, or 𝑥 > 6. Adding 6 and 12 in the inequality 6 + 12 > 𝑥 yields 18 > 𝑥, or 𝑥 < 18. Subtracting 12 from both sides of the inequality 12 + 𝑥 > 6 yields 𝑥 > 6 - 12, or 𝑥 > - 6. Since all x-values that satisfy the inequality 𝑥 > 6 also satisfy the inequality 𝑥 > - 6, it follows that the inequalities 𝑥 > 6 and 𝑥 < 18 represent the possible values of 𝑥. Therefore, the inequality 6 < 𝑥 < 18 represents the possible lengths, 𝑥, of the third side of the triangle. Choice A is incorrect. This inequality gives the upper bound for 𝑥 but does not include its lower bound. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID 90095507 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: 90095507 Townsend Realty Group Investments Property Purchase price Monthly rental address (dollars) price (dollars) Clearwater 128,000 950 Lane Driftwood 176,000 1,310 Drive Edgemont 70,000 515 Street Glenview 140,000 1,040 Street Hamilton 450,000 3,365 Circle The Townsend Realty Group invested in the five different properties listed in the table above. The table shows the amount, in dollars, the company paid for each property and the corresponding monthly rental price, in dollars, the company charges for the property at each of the five locations. Townsend Realty purchased the Glenview Street property and received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Which of the following best approximates the original price, in dollars, of the Glenview Street property? A. $350,000 B. $291,700 C. $233,300 D. $175,000 ID: 90095507 Answer Correct Answer: B Rationale Choice B is correct. Let x be the original price, in dollars, of the Glenview Street property. After the 40% discount, the price of the property became dollars, and after the additional 20% off the discounted price, the price of the property became. Thus, in terms of the original price of the property, x, the purchase price of the property is. It follows that. Solving this equation for x gives. Therefore, of the given choices, $291,700 best approximates the original price of the Glenview Street property. Choice A is incorrect because it is the result of dividing the purchase price of the property by 0.4, as though the purchase price were 40% of the original price. Choice C is incorrect because it is the closest to dividing the purchase price of the property by 0.6, as though the purchase price were 60% of the original price. Choice D is incorrect because it is the result of dividing the purchase price of the property by 0.8, as though the purchase price were 80% of the original price. Question Difficulty: Hard Question ID 98d3393a Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in two variables ID: 98d3393a Line in the xy-plane is perpendicular to the line with equation. What is the slope of line ? A. 0 B. C. D. The slope of line is undefined. ID: 98d3393a Answer Correct Answer: A Rationale Choice A is correct. It is given that line is perpendicular to a line whose equation is x = 2. A line whose equation is a constant value of x is vertical, so must therefore be horizontal. Horizontal lines have a slope of 0, so has a slope of 0. Choice B is incorrect. A line with slope is perpendicular to a line with slope 2. However, the line with equation x = 2 is vertical and has undefined slope (not slope of 2). Choice C is incorrect. A line with slope –2 is perpendicular to a line with slope. However, the line with equation x = 2 has undefined slope (not slope of ). Choice D is incorrect; this is the slope of the line x = 2 itself, not the slope of a line perpendicular to it. Question Difficulty: Hard Question ID 6989c80a Assessment Test Domain Skill Difficulty SAT Math Algebra Linear functions ID: 6989c80a The function gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of kelvins. If a temperature increased by kelvins, by how much did the temperature increase, in degrees Fahrenheit? A. B. C. D. ID: 6989c80a Answer Correct Answer: A Rationale 9 Choice A is correct. It’s given that the function 𝐹𝑥 = 𝑥 - 273.15 + 32 gives the temperature, in degrees Fahrenheit, that 5 corresponds to a temperature of 𝑥 kelvins. A temperature that increased by 2.10 kelvins means that the value of 𝑥 increased 9 by 2.10 kelvins. It follows that an increase in 𝑥 by 2.10 increases 𝐹 ( 𝑥 ) by 2.10, or 3.78. Therefore, if a temperature increased 5 by 2.10 kelvins, the temperature increased by 3.78 degrees Fahrenheit. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID 0cb57740 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear equations in one variable ID: 0cb57740 Each side of a -sided polygon has one of three lengths. The number of sides with length is times the number of sides with length. There are sides with length. Which equation must be true for the value of ? A. B. C. D. ID: 0cb57740 Answer Correct Answer: B Rationale Choice B is correct. It’s given that each side of a 30-sided polygon has one of three lengths. It's also given that the number of sides with length 8 centimeters cm is 5 times the number of sides 𝑛 with length 3 cm. Therefore, there are 5 × 𝑛, or 5𝑛, sides with length 8 cm. It’s also given that there are 6 sides with length 4 cm. Therefore, the number of 3 cm, 4 cm, and 8 cm sides are 𝑛, 6, and 5𝑛, respectively. Since there are a total of 30 sides, the equation 𝑛 + 6 + 5𝑛 = 30 represents this situation. Combining like terms on the left-hand side of this equation yields 6𝑛 + 6 = 30. Therefore, the equation that must be true for the value of 𝑛 is 6𝑛 + 6 = 30. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors. Question Difficulty: Hard Question ID e8f9e117 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: e8f9e117 The formula above is Ohm’s law for an electric circuit with current I, in amperes, potential difference V, in volts, and resistance R, in ohms. A circuit has a resistance of 500 ohms, and its potential difference will be generated by n six-volt batteries that produce a total potential difference of volts. If the circuit is to have a current of no more than 0.25 ampere, what is the greatest number, n, of six-volt batteries that can be used? ID: e8f9e117 Answer Rationale The correct answer is 20. For the given circuit, the resistance R is 500 ohms, and the total potential difference V generated by n batteries is volts. It’s also given that the circuit is to have a current of no more than 0.25 ampere, which can be expressed as. Since Ohm’s law says that , the given values for V and R can be substituted for I in this inequality, which yields. Multiplying both sides of this inequality by 500 yields , and dividing both sides of this inequality by 6 yields. Since the number of batteries must be a whole number less than 20.833, the greatest number of batteries that can be used in this circuit is 20. Question Difficulty: Hard Question ID d8539e09 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: d8539e09 For which of the following tables are all the values of and their corresponding values of solutions to the given inequality? A. B. C. D. ID: d8539e09 Answer Correct Answer: C Rationale Choice C is correct. All the tables in the choices have the same three values of 𝑥, so each of the three values of 𝑥 can be substituted in the given inequality to compare the corresponding values of 𝑦 in each of the tables. Substituting 3 for 𝑥 in the given inequality yields 𝑦 < 63 + 2, or 𝑦 < 20. Therefore, when 𝑥 = 3, the corresponding value of 𝑦 is less than 20. Substituting 5 for 𝑥 in the given inequality yields 𝑦 < 65 + 2, or 𝑦 < 32. Therefore, when 𝑥 = 5, the corresponding value of 𝑦 is less than 32. Substituting 7 for 𝑥 in the given inequality yields 𝑦 < 67 + 2, or 𝑦 < 44. Therefore, when 𝑥 = 7, the corresponding value of 𝑦 is less than 44. For the table in choice C, when 𝑥 = 3, the corresponding value of 𝑦 is 16, which is less than 20; when 𝑥 = 5, the corresponding value of 𝑦 is 28, which is less than 32; when 𝑥 = 7, the corresponding value of 𝑦 is 40, which is less than 44. Therefore, the table in choice C gives values of 𝑥 and their corresponding values of 𝑦 that are all solutions to the given inequality. Choice A is incorrect. In the table for choice A, when 𝑥 = 3, the corresponding value of 𝑦 is 20, which is not less than 20; when 𝑥 = 5, the corresponding value of 𝑦 is 32, which is not less than 32; when 𝑥 = 7, the corresponding value of 𝑦 is 44, which is not less than 44. Choice B is incorrect. In the table for choice B, when 𝑥 = 5, the corresponding value of 𝑦 is 36, which is not less than 32. Choice D is incorrect. In the table for choice D, when 𝑥 = 3, the corresponding value of 𝑦 is 24, which is not less than 20; when 𝑥 = 5, the corresponding value of 𝑦 is 36, which is not less than 32; when 𝑥 = 7, the corresponding value of 𝑦 is 48, which is not less than 44. Question Difficulty: Hard Question ID 48fb34c8 Assessment Test Domain Skill Difficulty SAT Math Algebra Linear inequalities in one or two variables ID: 48fb34c8 For which of the following tables are all the values of and their corresponding values of solutions to the given inequality? A. B. C. D. ID: 48fb34c8 Answer Correct Answer: D Rationale Choice D is correct. All the tables in the choices have the same three values of 𝑥, so each of the three values of 𝑥 can be substituted in the given inequality to compare the corresponding values of 𝑦 in each of the tables. Substituting 3 for 𝑥 in the given inequality yields 𝑦 > 133 - 18, or 𝑦 > 21. Therefore, when 𝑥 = 3, the corresponding value of 𝑦 is greater than 21. Substituting 5 for 𝑥 in the given inequality yields 𝑦 > 135 - 18, or 𝑦 > 47. Therefore, when 𝑥 = 5, the corresponding value of 𝑦 is greater than 47. Substituting 8 for 𝑥 in the given inequality yields 𝑦 > 138 - 18, or 𝑦 > 86. Therefore, when 𝑥 = 8, the corresponding value of 𝑦 is greater than 86. For the table in choice D, when 𝑥 = 3, the corresponding value of 𝑦 is 26, which is greater than 21; when 𝑥 = 5, the corresponding value of 𝑦 is 52, which is greater than 47; when 𝑥 = 8, the corresponding value of 𝑦 is 91, which is greater than 86. Therefore, the table in choice D gives values of 𝑥 and their corresponding values of 𝑦 that are all solutions to the given inequality. Choice A is incorrect. In the table for choice A, when 𝑥 = 3, the corresponding value of 𝑦 is 21, which is not greater than 21; when 𝑥 = 5, the corresponding value of 𝑦 is 47, which is not greater than 47; when 𝑥 = 8, the corresponding value of 𝑦 is 86, which is not greater than 86. Choice B is incorrect. In the table for choice B, when 𝑥 = 5, the corresponding value of 𝑦 is 42, which is not greater than 47; when 𝑥 = 8, the corresponding value of 𝑦 is 86, which is not greater than 86. Choice C is incorrect. In the table for choice C, when 𝑥 = 3, the corresponding value of 𝑦 is 16, which is not greater than 21; when 𝑥 = 5, the corresponding value of 𝑦 is 42, which is not greater than 47; when 𝑥 = 8, the corresponding value of 𝑦 is 81, which is not greater than 86. Question Difficulty: Hard Question ID f718c9cf Assessment Test Domain Skill Difficulty SAT Math Algebra Systems of two linear equations in two variables ID: f718c9cf The solution to the given system of equations is. What is the value of ? ID: f718c9cf Answer Correct Answer: 1.8, 9/5 Rationale 9 The correct answer is. Multiplying the first equation in the given system by 2 yields 10𝑥 + 28𝑦 = 90. Subtracting the second 5 equation in the given system, 10𝑥 + 7𝑦 =