Algebra 1 Section 6 PDF

LES SON Simplifying Rational Expressions with Like 51 Denominators Warm Up 1. Vocabulary _ (39) 2 3x is a(n)...

LES SON Simplifying Rational Expressions with Like 51 Denominators Warm Up 1. Vocabulary _ (39) 2 3x is a(n) expression. Simplify. 2. 5n - k + n (18) Factor each expression using the GCF. 3. 4x2 + x 4. w2 - w (38) (38) 5. Multiple Choice Which expression is equivalent to 6g2 - 12g + 3? (38) A 3(g - 5) B 6g(g - 2) + 1 C 3g(2g2 - 4g + 1) D 3(2g2 - 4g + 1) New Concepts Rational expressions have variables in the denominator. All values that would make the denominator equal zero are excluded. Example 1 Identifying Excluded Values Math Reasoning Find the excluded value(s) in each expression. Analyze Compare substituting a variable a. _ 3 that makes the 5m denominator equal to zero and dividing by SOLUTION zero. 5m = 0 Set the denominator equal to zero. _5 m = _0 Divide both sides by 5. 5 5 m=0 m≠0 0 is an excluded value. _5m b. m + 1 SOLUTION m+1=0 Set the denominator equal to zero. -1 = __ __ -1 Subtract 1 from both sides. Online Connection m = -1 www.SaxonMathResources.com m ≠ -1 -1 is an excluded value. 322 Saxon Algebra 1  Example 2 Simplifying Rational Expressions Simplify each rational expression, if possible. Identify any excluded values. a. _ 8h2 12h SOLUTION 12h = 0 Set the denominator equal to zero. h=0 Solve. h≠0 0 is the excluded value. _ 8h2 12h 1 4h(2h) =_ Factor out the GCF. 4h(3) 1 =_ 2h Simplify. 3 b. _ 2m - 6 m-3 SOLUTION m -3 = 0 Set the denominator equal to zero. m=3 Solve. m≠3 3 is the excluded value. _ 2m - 6 m-3 1 2 (m - _ 3) Factor the numerator. Divide out common factors. (m - 3) 1 =2 Simplify. 2 4y + 8y _ Hint c. y+2 If there is more than one term in the numerator or SOLUTION denominator, begin by factoring. y+2=0 Set the denominator equal to zero. y + 2 cannot be factored. y = -2 Solve. 4y2 + 8y can be factored. y ≠ -2 -2 is the excluded value. 2 4y + 8y _ y+2 1 4y(y + 2) =_ Factor the numerator. Divide out common factors. (y + 2) 1 = 4y Simplify. Lesson 51 323  Example 3 Simplifying Expressions with Integer Exponents Simplify each rational expression, if possible. a. _ b -_ 2b a3 a3 SOLUTION _ b -_ 2b a3 a3 = -_ b The expressions have like denominators. Subtract a3 the numerators. b. pq-2 - _ Hint 7 p p-1q2 pq-2 is the same as _ q2. SOLUTION 7p pq-2 - _2 q p 7p = _2 - _2 Write the expression with only positive exponents. q q 6p = -_2 The expressions have like denominators. Subtract the q numerators. Example 4 Application: Fencing a Field A rectangular field has length _ 6a 3a + 4 _ 6a km 3a + 4 kilometers and width _ 8 3a + 4 kilometers. Find the amount of fencing needed to _ 8 km enclose all four sides. 3a + 4 SOLUTION P = 2(l + w) Write the formula for the perimeter of a rectangle. P=2 (_ 6a + _ 3a + 4 3a + 4) 8 Substitute for the length and width. = 2(_) 6a + 8 The expressions have like denominators. Add the 3a + 4 numerators. 1 =2 ( 2(3a + 4) _ 3a + 4 1 ) Factor the numerator. The GCF is 2. = 2(2) Simplify. =4 4 kilometers of fencing is needed. 324 Saxon Algebra 1  Lesson Practice Find the excluded values in each expression. (Ex 1) a. _ 9 4h p+2 b. _ p+4 g-5 _ c. 3g - 15 Simplify each rational expression, if possible. Identify any excluded values. (Ex 2) d. _ 4a3 e. _ d+1 2a2 d _ 3z2 - 6z 5xy − 10x _ f. g. 5z - 10 x2y2 Simplify each rational expression, if possible. (Ex 3) 4f _ _ 2f h. - r2 r2 i. 6m-2n4 + _ -2 3m n-4 j. Framing A rectangular mirror has a frame with length _ 2x + 5 (Ex 4) 4y inches and _ 6-x width 4y inches. Find the perimeter of the frame. _ 2x + 5 in. 4y _ 6 x in. 4y Practice Distributed and Integrated Simplify. 4 3 1. √81 2. √ -27 (46) (46) 3 3 3. √ 64 4. √ -64 (46) (46) 5. Write 7y = _8 x - 1 in standard form. 3 (35) 6. Simplify _. 12x2 - 16x (38) 16xy Lesson 51 325 Add. *7. _d m-2 + _ 5d (51) 3 m2 y-2 *8. _8h-6 + _ (51) y2 h6 *9. Multiple Choice What is the excluded value for _ h+3 (51) 2h - 6 ? A h ≠ -2 B h≠0 C h≠2 D h≠3 160 + 160f *10. Football A NCAA football field has a width of _ f+1 feet and a length (51) 360 + 360f of _ feet. A person walks the outside boundary line to mark it with chalk. f+1 How far does the person walk? __ 360 + 360f ft f +1 __ 160 + 160f ft f+1 5p *11. Write Find the excluded value of the rational expression _ p - 6. Explain. (51) 12. Travel You begin a long-distance trip along the highway at mile marker 21. (34) After 5 minutes, you pass mile marker 32. After another 5 minutes, you pass mile marker 43. You have been traveling at a constant rate. If you continue to travel at this constant rate, what mile marker will you be at after 60 minutes? (Hint: Consider marker 21 (after 0 minutes) to be a1, marker 32 (5 minutes) to be a2, marker 43 (after 10 minutes) to be a3, and so on.) 13. Write the converse of the following statement: If a number is a whole number, (Inv 5) then the number is a natural number. Then determine whether the new statement is true or false. If false, give a counterexample. 14. Multiple Choice The following data set shows the height in inches of 9 eighth (48) graders: 66, 62, 56, 64, 60, 62, 58, 57, 59. What is the range of the data? A7 B 10 C 60 D 62 15. The scale factor of two similar triangles is 4:5. If one angle of the smaller triangle (36) measures 60°, what is the measure of its corresponding angle in the larger triangle? 10 8 5 √3 4 √3 60° 4 5 326 Saxon Algebra 1  16. Produce Cost Find the unit price of apples at the grocery store. (41) Apples (lb) 1.5 2 3 3.5 Cost $2.40 $3.20 $4.80 $5.60 *17. Multi-Step Luca and Paolo buy old bikes for a few dollars each, and then fix them (42) up so that they can sell them for a profit. If they bought a rare tandem bike for $8, spent an additional $150 on the bike for paint and parts, and sold it at a 285% profit, then how much money did they sell the bike for? a. Analyze If profit means the amount of money Luca and Paolo made after taking into account what they spent, write an equation to find how much the bike sold for. b. Solve the equation to find the sale price of the bike. *18. Write If a rational expression has values at which the numerator equals 0, are (43) these points undefined for the expression? Explain. 19. Determine the slope of the line shown on the graph. (44) y (-2, 6) x -8 -4 (5, -3) -4 -8 20. Find the percent of increase or decrease if the original price was $7000 and the (47) new price is $10,200. 21. Analyze Determine if the premise and the conclusion use inductive or deductive (Inv 4) reasoning. Explain your choice. Premise: The measures of two angles in a triangle add up to 80°. Conclusion: The third angle is 100°. *22. Astronomy The surface temperature of the sun is about 5880 kelvins. The (50) sun’s core is much hotter. Create a graph of an inequality that represents the temperature of the sun’s core. *23. Geometry According to a city bylaw, the area of a store’s sign cannot exceed (50) 900 square decimeters. Write and graph an inequality to represent the situation. 24. Justify The numbers of students who bought lunch each day this month are (48) listed below. 176, 134, 208, 170, 149, 153, 136, 200, 168, 150, 157, 141, 211, 176, 145, 155, 128, 199, 182, 148. a. How would you find the median of the set of data above? b. Find the median. Lesson 51 327  25. Write the equation 5x + 3y = 9 in slope-intercept form. (49) 26. Multiple Choice Which equation has a slope of -_12 and a y-intercept of -3? (49) A 4x + 2y + 3 = 0 B 3x + 6y + 6 = 0 C 5x + 10y + 30 = 0 D 6x + 2y + 1 = 0 *27. Determine which values in the set {-1, 0, 1, 2} are solutions to the inequality (50) x - 1 ≥ 4. 28. Error Analysis Two students graphed the statement “A number is at least 9” as shown (50) below. Which student is correct? Explain the error. Student A Student B 6 8 10 12 6 8 10 12 29. Multi-Step The graph shows that the temperature is now at least the temperature it (50) was this morning. How can the current temperature be expressed with a sentence? -18 -16 -14 -12 a. Write an inequality for the graph. b. Translate the inequality into a sentence. 3 4 30. Measurement The specifications for a housing development include that each lot has to be at least 1_2 acres. Write and graph an inequality to represent the (50) 1 situation. 328 Saxon Algebra 1 LESSON Determining the Equation of a Line 52 Given Two Points Warm Up 1. Vocabulary The (41) is a measure of the steepness of a line. 2. Find the x-intercept in the equation 3x + y = 6. (35) 3. Find the y-intercept in the equation 3x + y = 6. (35) Solve each proportion. 4. _ 2 =_ 4 n 5. _ 5 =_ 1 (36)9 (36) x + 10 3 New Concepts A line can be graphed if the slope and any point y on the line are known. For example, to graph a 4 (2, 4) line that has a slope of -_23 and passes through 2 (2, 4), begin by plotting the point (2, 4). From (5, 2) O x the point (2, 4), count down two units and to the -4 -2 2 4 right three units. Plot a point there. Then draw -2 a line through the two points (2, 4) and (5, 2). -4 Example 1 Using Slope and a Point to Graph a. Graph a line that has a slope of 4 and passes through point (3, 5). Hint SOLUTION Graph the point (3, 5). y A slope of 4 is the same (4, 9) as _41. 8 From the point (3, 5), count up four units and to the right one unit. Graph a point there. 6 Sketch the line through the two points (3, 5) 4 (3, 5) and (4, 9). 2 Math Language x The slope of a line 2 4 6 8 represents a rate of change. A positive slope slants up to the right. b. Graph a line that has a slope of 0 and passes through point (2, 3). A negative slope slants down to the right. SOLUTION y 4 (2, 3) Graph the point (2, 3). 2 A line with a slope of 0 is a horizontal line. O x -4 -2 2 4 Online Connection Sketch a horizontal line that passes through -2 www.SaxonMathResources.com the point (2, 3). -4 Lesson 52 329  Point-Slope Form The form y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line is called the point-slope form of a line. Example 2 Writing an Equation in Point-Slope Form Write the equation of a line that has a slope of 3 and passes through point (2, 4) in point-slope form. SOLUTION y - y1 = m(x - x1) Write the formula. y - 4 = 3(x - 2) Substitute (2, 4) for (x1, y1) and 3 for m. Example 3 Writing an Equation Using Two Points Write the an equation of a line that passes through the points (1, -3) and (4, 5) in slope-intercept form. SOLUTION y2 - y1 m=_ x -x Write the slope formula. 2 1 5 - (-3) m=_ Substitute (1, -3) for (x1, y1) and (4, 5) for (x2, y2). 4-1 =_ 8 Simplify. 3 The slope is = _ 8. 3 Hint y - y1 = m(x - x1) Write the point-slope formula. You can substitute either y-5=_ 8 (x - 4) Substitute (4, 5) for (x1, y1) and _ 8 for m. (1, -3) or (4, 5) for (x1, y1). 3 3 3(y - 5) = 3 · _ 8 (x - 4) Multiply both sides by 3. 3 3y - 15 = 8x - 32 Distribute the 3 and 8. y=_ 8x - _ 17 Solve for y. 3 3 Check y (4, 5) 4 Graph y = _8x - _17. 3 3 2 The graph of y = _8 x - _ 3 17 appears to go 3 O x through the points (1, -3) and (4, 5). -4 -2 4 -2 (1, -3) -4 330 Saxon Algebra 1  Example 4 Application: Sales Rachel is selling seashells. She sold 1 for $5 and 2 for $8. What will Rachel charge for 5 seashells if she keeps selling the seashells at the same price rate? Math Reasoning Understand Predict Suppose Rachel The ordered pair (1, 5) represents “1 shell for $5.” sells some shells and receives $200. How many The ordered pair (2, 8) represents “2 shells for $8.” shells would she have sold if she had continued The slope represents the price rate, which is constant. to sell the shells at the same price rate? Plan Find the slope of the line. Write an equation of the line that passes through (1, 5) and (2, 8). Use the equation to find the value of y when x = 5. Solve y2 - y1 m=_ x -x Write the slope formula. 2 1 =_ 8-5 Substitute (1, 5) for (x1, y1) and (2, 8) for (x2, y2). 2-1 =3 Simplify. y - y1 = m(x - x1) Write the point-slope formula. y - 5 = 3(x - 1) Substitute (1, 5) for (x1, y1) and 3 for m. y - 5 = 3x - 3 Distribute the 3. y = 3x + 2 Solve for y. y = 3(5) + 2 Substitute 5 for x. y = 17 y (5, 17) 16 Rachel will sell 5 seashells for $17. 12 Check 8 (2, 8) Graph y = 3x + 2. (1, 5) 4 The graph representing y = 3x + 2 goes through the O x points (1, 5), (2, 8), and (5, 17). 2 4 6 8 Lesson Practice a. Graph a line that has a slope of 2 and passes through the point (5, 6). (Ex 1) b. Graph a line that has a slope of 0 and passes through the point (-1, 1). (Ex 1) c. Write the equation of a line that has a slope of 6 and passes through the (Ex 2) point (7, 9) in point-slope form. d. Write the equation of a line that passes through the points (2, -3) and (Ex 3) (7, 4) in slope-intercept form. Lesson 52 331  e. Trevor began a computer game with 3 points. After 1 minute he had (Ex 4) -1 points, and after 2 minutes he had -5 points. How many points will he have after 3 minutes of playing the computer game if he continues losing at the same rate? Practice Distributed and Integrated Given the domain of a function, find the range. 1. f (x) = 3x - 5; domain: {0, 1, 2, 3} (25) 2. f (x) = _ 1 x + 3; domain: {-2, 0, 2, 4} (25) 2 Write an inequality for each situation. 3. To qualify for the job, the applicants must have more than 3 years of experience in (45) the field. 4. In 2005 the minimum wage in the United States was $5.15 per hour. (45) (3 × 10 -9 ) 5. Divide _-1 and write the answer in scientific notation. (37) (4.8 × 10 ) *6. Graph the line that has a slope of -1 and passes through the point (3, 1). (52) 7. The scale of a map is 1 in.:20 mi. Find the actual distance that (36) corresponds to a map distance of 4 inches. 8. Carpentry You are constructing a picnic table and are using a scale of 1 inch to (36) 6 inches. If the length of the table on the drawing is 7 inches, what will the actual length of the table be? 9. A box with a volume of _ (39) t3 _ 4 t2 2 y 5y +_ ( ) m receives a fixed postage rate. y a. Simplify the expression above. b. Identify the variables that cannot equal zero. 10. Simple Interest Simple interest is the amount of money the borrower pays based on (42) the amount borrowed (the principal) for a given period of time (months or years). It is calculated this way: I = prt. If a person borrows $20,000 (p) to buy a car, pays 6.95% interest (r), and takes 5 years (t) to repay the loan, how much will the borrower pay in simple interest? 11. Analyze The rational expression _ x-5 15x2 - 75x is not defined for all real numbers. (43) a. When is the denominator equal to 0? b. When is the numerator equal to 0? c. When is the rational expression undefined? 12. Analyze The slope of a line is 3. Two points on this line are (-1, -2) and (4, ?). Using (44) the formula for slope, determine the missing y-value for the second point. 332 Saxon Algebra 1  13. Translate the following sentence into an inequality: (45) The difference of -4 and an unknown number is less than or equal to 0. 14. Evaluate the expression for the given value. (46) - √x  when x = _ 25 16 *15. Business Arminda owns a ladies’ bag business. The total cost of producing a (28) group of bags is $1100. In addition, each bag costs $18 in materials. If Arminda sells each bag for $40, how many bags should Arminda sell to gain a profit of at least $2200? 16. The data below are the weights of 10 newborns (to the nearest pound). Find the (48) mean, median, and mode of the data. 5, 8, 6, 5, 7, 8, 10, 7, 8, 6. *17. Data Analysis Study the graph. It shows the cost of playing Cost of Playing (49) x number of games after renting equipment. a. Write an equation that represents the line of the graph 70 60 in slope-intercept form. Cost ($) 50 40 b. What is the cost of playing 5 games? 30 20 10 0 1 2 3 4 5 6 Number of Games 18. Multi-Step Murietta is planning a party at a bowling alley. She wants to rent two (49) lanes. The rental fee for the two lanes is $40 plus the rental of $2 per pair of bowling shoes. a. Write an equation in slope-intercept form to represent this situation. b. Murietta is renting two lanes and inviting 9 people. If everyone including herself rents shoes, what will the cost of the party be? 19. Write an inequality for the graph below. (50) 0 1 *20. Error Analysis Two students graphed the inequality 2.5 > b as shown below. (50) Which student is correct? Explain the error. Student A Student B 1 2 3 4 1 2 3 4 21. Find the excluded value in the expression _3. 5 b (51) b *22. Soccer In international play, the maximum length of a soccer field is _ 525x + 100 5x + 1 (51) _ 400x + 85 meters and the maximum width is 5x + 1 meters. Each linesman is responsible for watching half of the field’s perimeter. How many meters along the border of the field must the linesman watch during the game? Lesson 52 333 *23. Write the inverse of the statement: “If a polygon is has four sides, then it is a (Inv 5) quadrilateral.” Then determine whether the new statement is true or false. If false, give a counterexample. 24. Error Analysis Two students simplify the expression -r3s-4 + _ -4 2s. Which student is (51) r -3 correct? Explain the error. Student A Student B -r3s-4 + _ -r3s-4 + _ -4 -4 2s 2s r-3 r-3 =_ -r3 _ 3 cannot simplify 4 + 2r4 s s = _4 3 r s 25. Geometry Two sides of a triangle measure 4x + 8 and x2 + 2x. Find the (43) ratio of the first side to the second side. Simplify, if possible. 2 x + 2x 4x + 8 26. A rectangular window has a length of _ 6a 3a - 1 meters and a width of _ 4 3a - 1 meters. (51) How many feet of trim will be painted if the trim goes around three sides of the window (both lengths and one width)? *27. Multiple Choice On the first day of the school play, Carlos sold four tickets. On the (52) second day, he sold seven tickets. Which equation of a line represents the line that passes through the two points that represent the data in this problem? A y = 3x + 1 B y=_ 1x + 1 C y = -3x + 1 D y = 3x - 1 3 *28. Write Find the slope of the line that passes through (-1, 2) and (3, 2). (52) Explain. *29. Rachel and a Michelle are crocheting a baby blanket that will be 72 inches long. (52) Rachel crochets the first 24 inches and then gives the blanket to Michelle to finish. Michelle expects to crochet at a rate of 8 inches per day. How many days will it take Michelle to finish the blanket? a. Write an equation giving the length y of the blanket (in inches) when crocheting for x days. b. Graph the equation using a graphing calculator. c. How long it will take Michelle to finish the blanket? *30. Transportation John is 500 miles from home. He is traveling toward home at a (52) constant rate of 65 miles per hour. The distance d (in miles) away from home after t (in hours) is given by the equation d = 500 - 65t. a. Use a graphing calculator to make a table of values with the values of t from 0 to 5 in increments of 1. b. How far away from home John is after 4 hours? 334 Saxon Algebra 1 LESSON Adding and Subtracting Polynomials 53 Warm Up 1. Vocabulary A part of an expression that is added to or subtracted from (2) the other parts is called a. Simplify. (_49 ) 0 2. (32) 3. (_ ) -1 12 (32) 8 4. Find the GCF of 25b3, 50b2, and 100b. (38) New Concepts A monomial is the product of numbers and/or variables with whole-number exponents. Monomials Not Monomials x, _ x3 , -xy, 0.75x2, 3 _1 , _ 2 , x - y, 0.75x-2 2 x x3 The degree of a monomial is the sum of the exponents of the variables in the monomial. A constant has a degree of 0. Example 1 Finding the Degree of Monomials Find the degree of each monomial. Caution a. -7x2yz3 Remember y = y1 SOLUTION -7x2yz3 Find the sum of the exponents of the variables. 2+1+3=6 The degree of the monomial is 6. b. 8xy2z SOLUTION 8xy2z Find the sum of the exponents of the variables. 1+2+1=4 The degree of the monomial is 4. Caution c. 122ab3c Be careful and do not SOLUTION include the exponent on the base 12. Use 122ab3c Find the sum of the exponents of the variables. only the exponents of the variables to find the 1+3+1=5 degree. The degree of the monomial is 5. Lesson 53 335  A polynomial is a monomial or the sum or difference of monomials. Polynomials A polynomial with one term is a monomial. 6x A polynomial with two terms is a binomial. 6x + 10 A polynomial with three terms is a trinomial. x2 + 6x + 10 The degree of a polynomial is the degree of the greatest-degree term in the polynomial. The leading coefficient for a polynomial is the coefficient of the term with the greatest degree. The standard form of a polynomial is a form of a polynomial where terms are ordered from greatest to least degree. Example 2 Writing a Polynomial in Standard Form Write each polynomial in standard form. Then find the leading coefficient. a. 2n2 + n3 SOLUTION 2n2 : degree 2 n3 : degree 3 n3 + 2n2 is in standard form. The leading coefficient is 1. b. 8xy2 - 9 + 5x3y3z SOLUTION 8xy2 Add the exponents of the variables: 1 + 2 = 3. 5x3y3z Add the exponents of the variables: 3 + 3 + 1 = 7. -9 A constant has a degree of 0. 5x3y3z + 8xy2 - 9 Arrange the terms in descending order. 7 3 0 5x3y3z + 8xy2 - 9 is in standard form. The leading coefficient is 5. c. 9x2y - 3x2y2 - 5xy SOLUTION 9x2y Add the exponents of the variables: 2 + 1 = 3. -3x2y2 Add the exponents of the variables: 2 + 2 = 4. 5xy Add the exponents of the variables: 1 + 1 = 2. -3x2y2 - 9x2y - 5xy Arrange the terms in descending order. 4 3 2 Online Connection 2 2 2 www.SaxonMathResources.com -3x y + 9x y - 5xy is in standard form. The leading coefficient is -3. 336 Saxon Algebra 1  Exploration Using Algebra Tiles to Add or Subtract Polynomials Use algebra tiles to find (x2 + 3x - 3) + (2x - x + 2). Materials a. Model each expression using algebra tiles. algebra tiles (x2 + 3x - 3) (2x2 - x + 2) - + + + + + + - - + - + b. Rearrange tiles so that like terms are together. - + + + + - + - + + - + c. Remove the zero pairs. - + + + + - + - + + - + d. The remaining tiles represent the sum. - + + + + + The sum is 3x2 + 2x - 1. e. Model Use algebra tiles to find (2x2 + 3x) + (2x - x2) - x2. Lesson 53 337  To add or subtract polynomials, combine like terms. Polynomials can be added vertically or horizontally. Example 3 Adding Polynomials Add the polynomials. Write the answer in standard form. a. (-8x3 + 4x2 + x + 1) + (3x3 - 2x2 + 7) SOLUTION -8x3 + 4x2 + x + 1 Arrange like terms in columns. +3x3 - 2x2 +7 -5x3 + 2x2 + x + 8 Add like terms. The polynomial -5x3 + 2x2 + x + 8 is in standard form. b. (x + 10 + 2x3) + (3x2 - 7x - 2) Math Language SOLUTION Like terms, such as -8x3 (x + 10 + 2x3) + (3x2 - 7x - 2) and 3x3, have the same variables raised to the x + 10 + 2x3 + 3x2 - 7x - 2 Remove parentheses. same powers. 2x3 + 3x2 + x - 7x - 2 + 10 Arrange with like terms together. 3 2 2x + 3x - 6x + 8 Combine like terms. The polynomial 2x3 + 3x2 - 6x + 8 is in standard form. Example 4 Subtracting Polynomials Caution Subtract the polynomials. Write the answer in standard form. Remember to multiply a. (6x2 + 4x + 2) - (2x2 - x + 8) each term in the parentheses by -1. SOLUTION 2 -1(2x - x + 8). (6x2 + 4x + 2) - (2x2 - x + 8) -2x2 + x - 8 Find the opposite of the second polynomial. 6x2 + 4x + 2 Arrange like terms in columns. 2 + -2x + x - 8 4x2 + 5x - 6 Add like terms. The polynomial 4x2 + 5x - 6 is in standard form. 338 Saxon Algebra 1 Math Reasoning b. (x2 + 4x - 9) - (4x2 - 5x + 11) Generalize When a SOLUTION polynomial is in standard form, is the leading (x2 + 4x - 9) - (4x2 - 5x + 11) coefficient always the coefficient of the first x2 + 4x - 9 - 4x2 + 5x - 11 Remove the parentheses. term? x2 - 4x2 + 4x + 5x - 9 - 11 Arrange like terms together. 2 -3x + 9x - 20 Combine like terms. The polynomial -3x2 + 9x - 20 is in standard form. Example 5 Application: Internet For the years 1995 through 2001, the number of websites for businesses and education can be represented by the expressions below. business websites: 0.333t2 - 1.035t + 0.607 education websites: 0.098t2 - 0.121t + 0.296 Write an expression for the total number of business and educational websites. SOLUTION 0.333t2 - 1.035t + 0.607 + 0.098t2 - 0.121t + 0.296 0.431t2 - 1.156t + 0.903 The expression 0.431t2 - 1.156t + 0.903 represents the total number of websites. Lesson Practice Find the degree of each monomial. (Ex 1) a. 3x2yz6 b. -23xyz c. 42xy2z3 Write each polynomial in standard form. Then find the leading coefficient. (Ex 2) d. 3w2 - 2w4 e. 5ab2 + 3a2b2 + 8ab - 1 f. 2ab - 7 - 5a2b Add the polynomials. Write each answer in standard form. (Ex 3) g. (2x2 + x + 8) + (x2 + 4) h. (3n2 + 7n - 1) + (-2n2 - n + 1) Subtract the polynomials. Write each answer in standard form. (Ex 4) i. (12y3 + 10) - (18y3 - 3y2 + 5) j. (c2 + 6c - 2) - (c2 - 2c + 6) Lesson 53 339  k. Hector throws a baseball up into the air, and at the same time, Jonas (Ex 5) throws a another baseball upward. The expressions below represent the height of the baseballs at time t. Hector’s baseball: -16t2 + 22t + 4 Jonas’s baseball: -16t2 + 17t + 6 Write an expression for the difference in height between the two throws. Practice Distributed and Integrated Simplify each expression. 1. 18 - 12 + 42 (4) 2. -2[7 + 6(3 - 5)] (4) 3. Graph the inequality x ≤ 8. (50) 4. Write an inequality for the graph shown below. (50) 1 2 3 4 5. Compare: 0.00304 3.04 × 10-4. (37) 6. Factor the polynomial 18a2b3c - 45ab6c completely. (38) 7. Rockets The formula h = -16t2 + 80t + 8 can be used to find the height of a (38) rocket that is launched into the air from 4 feet off the ground with an initial velocity of 80 feet/second. Write the formula by factoring the right side of the equation using the GCF. 3 8. Simplify the expression (3a)(6a2b). (40) 9. Automotive Performance The ratio _40 + 6t 20 + 10t compares the speeds of a car starting (43) from two different cruising speeds after an acceleration lasting t seconds. What is the simplified form of the expression? 10. Multi-Step The inflation rate, the rate at which the things people buy and use (44) increase in price, is constantly changing. During the first month of 2007, the inflation rate was 2.08%. By the sixth month, the rate had changed to 2.69%. a. Graph this information on a coordinate plane as 2 points connected by a line segment. b. What is the slope of the line segment? c. Predict Use the slope to predict the probable inflation rate for July of 2007. 11. Write What phrases can be used to indicate the inequality ≥? (45) 12. Write the expression - √ b with a fractional exponent. (46) 340 Saxon Algebra 1  13. Astronomy The table shows the sidereal periods (the time it takes for a planet to (48) orbit the sun) for the 8 planets of our solar system. What is the mean time it takes a planet in our solar system to revolve around the sun? What is the median time? Planet Sidereal Period (in days) Mercury 88 Venus 225 Earth 365 Mars 687 Jupiter 4,329 Saturn 10,753 Uranus 30,660 Neptune 60,150 14. Multiple Choice Which of the following graphs is correct for -3 ≤ y? (50) A B -6 -4 -2 0 -6 -4 -2 0 C D -6 -4 -2 0 -4 -2 0 15. Generalize How can the graph of an inequality show if a number is in the (50) solution set? 16. Find the percent of increase or decrease to the nearest percent if the original price (47) was $48,763 and the new price is $39,400. 17. Identify the slope and y-intercept in the equation of a line given below. (49) 1.5x + 3y - 6 = 0 *18. Simplify the rational expression, if possible. Identify any excluded values. (51) _ 2k + 6 k+2 *19. Water Polo A water-polo pool is _ _ 50x + 150 40x (51) 3x + 5 meters long and 3x + 5 meters wide. Find the perimeter of the pool. y+4 20. Error Analysis Two students simplify the expression _ y - 2. Which student is correct? (51) Explain the error. Student A Student B y+4 _ y+4 _ y-2 y-2 cannot be simplified y+4 _ y-2 _4 -2 -2 Lesson 53 341  21. Geometry Points A, B, and C are three vertices of a rectangle. Plot the three (52) points. Then find the coordinates of the fourth point, D, to complete the rectangle. Finally, write the equation of the line that passes through points B and D and forms a diagonal of the rectangle. A (-2, 3), B (4, 3), C (4, -1) 3 4 *22. Measurement Two farmers each harvested 50 acres of tomatoes per day from (52) their fields. The area of one farmer’s field is 800 acres and the area of the other farmer’s field is 600 acres. a. Write an equation giving the unharvested area y of the larger field (in acres) after x days. b. Write an equation giving the unharvested area y of the smaller field (in acres) after x days. c. What is the unharvested area of each field (in acres) after 10 days? *23. Write an equation of a line that passes through points (6, -3) and has a slope (52) of -2. *24. Multiple Choice Which expression is not a polynomial? (53) A -12b B x2 + x-1 C y2 - y + 6 D -60 *25. Advertisement The polynomials below approximate the amount of dollars (in (53) millions) one company spent on advertising children’s books on the national and local level for each year during a certain period. In each polynomial, x represents the years since the company began its campaign. Write a polynomial that gives a combined amount spent each year on national and local advertising. national = 59x2 - 262x + 3888 local = -33x3 + 611x2 - 1433x + 28,060 *26. What is the degree of -a2b2c3 + 5x5? (53) *27. A line passes through the points (4, 5) and (6, 4). (52) a. Write the equation of the line in point-slope form. b. Write the equation of the line in slope-intercept form. c. Graph the line using a graphing calculator. d. Fill in the missing coordinates of the following points: (x, 4) and (3, y). 28. Write an equation of the line in point-slope form that has a slope of -1 (52) and passes through the point (3, 1). *29. Write Is 4 equal to 4x0? Explain. (53) *30. Verify What polynomial can be subtracted from 3x2 + 7x - 6 to get -6 ? (53) 342 Saxon Algebra 1  LAB 4 Drawing Box-and-Whisker Plots Graphing Calculator Lab (Use with Lesson 54) When making a box-and-whisker plot, remember to first place the given data in numeric order. Then separate that data into quartiles. Inadvertently omitting a value or listing the data out of order is a common error and will result in inaccurate quartiles. Use a graphing calculator to easily create a box-and-whisker plot based on a set of data, regardless of whether the data is given in numeric order. The data listed are the ages of 16 students’ oldest siblings. Make a box-and-whisker plot for the data. 10, 11, 15, 10, 9, 14, 12, 8, 9, 15, 7, 10, 10, 14, 18, 19 1. Press and choose 1:Edit to enter the data into L1 (List 1). 2. Clear any old data by pressing the key until L1 is selected and then by pressing and then. 3. Enter the data one at a time. Press after keying in each value. 4. Press and select 1:Plot1… to open the plot setup menu. 5. Press to turn Plot1 On and then press the key once and the key four times to select the icon in the middle of the bottom row. Online Connection The setting for Xlist should be L1 and Freq www.SaxonMathResources.com should be 1. Lab 4 343  6. Create a box-and-whisker plot by pressing and selecting 9:ZoomStat. 7. Use the key and then the and keys to view the statistical values. The minimum age is 7 years, the first quartile is 9.5 years, the median is 10.5 years, the third quartile is 14.5 years, and the maximum age is 19 years. Lab Practice a. Use the data below to make a box-and-whisker plot for the maximum speeds of these animals. Maximum Animal Speed (mph) Cheetah 70 Lion 50 Coyote 43 Hyena 40 Rabbit 35 Giraffe 32 Grizzly Bear 30 Cat 30 Elephant 25 Squirrel 12 The heights of 11 students in inches are as follows: 66, 60, 59, 67, 68, 63, 62, 61, 69, 64, and 61 b. Make a box-and-whisker plot of the data. c. What is the median height of the students? d. Between which heights do 50 percent of the students fall? 344 Saxon Algebra 1 LESSON Displaying Data in a Box-and-Whisker Plot 54 Warm Up 1. Vocabulary A(n) (48) is a data value that is much greater or much less than the other values in the data set. 2. Skateboards sell for $39.99, $32.99, $65.98, $38.99, $28.99, $31.00, $41.00. (48) Find the mean price of the skateboards. (Round the answer to the nearest dollar.) 3. Space Exploration The data set shows the number of days each expedition (48) crew was assigned to the International Space Station from 2000 to 2004. 140.98, 167.28, 128.86, 195.82, 184.93, 161.05, 184.93, 194.77, 185.66, 192.79 a. Find the median of the data set. b. Find the range of the data set. Evaluate. (8.2 + 3.7 + 9.1 + 3.8) 4. __ 5. -_ 7 ÷ -_ 3 ( ) (11) 4 (11) 8 4 New Concepts A box-and-whisker plot displays data that are divided into four groups. A line inside the box shows the median, the ends of the box show the quartiles, and the ends of the whiskers show the minimum and maximum values. In 15 hockey games, 1, 1, 1, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 9, 9 goals were scored. The number of goals scored can be displayed in a box-and-whisker plot. Hockey Goals Math Language First Third The median is the Quartile Quartile middle number in a set of numbers that are Minimum Median Maximum arranged from least to greatest. 0 1 2 3 4 5 6 7 8 9 10 Half of the games had between 3 to 8 goals per game. One-fourth of the games had between 1 and 3 goals per game. The median number of goals scored was 5. The greatest number of goals scored in one game was 9. The quartiles divide the data into fourths. The first quartile is the median of Online Connection the lower half of the data, and the third quartile is the median of the upper www.SaxonMathResources.com half of the data. Lesson 54 345  Determining Outliers The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). Outliers are any values x such that x < Q1 - 1.5(IQR) or x > Q3 + 1.5(IQR) Example 1 Analyzing a Box-and-Whisker Plot Math Language The box-and-whisker plot shows scores on Test Scores a history test. Use the interquartile range to An outlier is a data value that is much greater or identify outliers. much less than the other data values in the set. SOLUTION 50 60 70 80 90 100 first quartile (Q1): 70 third quartile (Q3): 90 Interquartile Range interquartile range (IQR): 90 - 70 = 20 → (1.5)20 = 30 Math Reasoning 70 - 30 = 40 90 + 30 = 120 Write In a box-and- No values are less than 40 or greater than 120, so there are no outliers. whisker plot, what do Q1, Q2, and Q3 represent in a data set? Example 2 Displaying Data in a Box-and-Whisker Plot Make a box-and-whisker plot to display the data of fish lengths (in millimeters) reported by the California Department of Fish and Game. 312, 210, 422, 323, 358, 511, 689, 722, 333, 301, 298, 755, 213, 245, 356 Half of the fish are between which lengths? SOLUTION Order the data from least to greatest. Then find the quartiles, median, minimum and maximum values. 210, 213, 245, 298, 301, 312, 323, 333, 356, 358, 422, 511, 689, 722, 755 minimum Q1 median Q3 maximum Draw a box-and-whisker plot. Length of Fish Half of the fish are between 298 mm and 511 mm. 200 300 400 500 600 700 800 In a box-and-whisker plot, outliers are represented by an asterisk (*) and are not included in the whisker. Example 3 Displaying Data Including Outliers The speeds of 10 birds were recorded in miles per hour. 19, 20, 22, 24, 24, 26, 29, 30, 32, 47 Display the data using a box-and-whisker plot. Identify any outliers. 346 Saxon Algebra 1  SOLUTION Find the median, Q1, Q3, and IQR. Identify any outliers. To find the median, calculate the mean of the two middle numbers. _ 24 + 26 = 25 2 Q1: 22 Q3: 30 IQR: 8 Q1 - 1.5(IQR) = 22 - 1.5(8) = 10 Q3 + 1.5(IQR) = 30 + 1.5(8) = 42 47 is an outlier because it is greater than 42. Make a box-and-whisker plot. The upper whisker will end at 32, and 47 will be represented by an asterisk. Flight Speeds of Birds 15 20 25 30 35 40 45 50 Example 4 Comparing Data Using a Box-and-Whisker Plot The table shows the average number of Rebounds Per Game, 2006–07 rebounds per game for the Texan Runners during the 2006–07 season. B. Gracia 10.6 T. Chin 3.2 a. Identify any outliers. M. Barry 4.4 SOLUTION J. Carl 2.7 R. Iffla 2.1 Q1 - 1.5(IQR) = 2.05 - 1.5(1.85) = -0.725 B. Abena 2.7 Q3 + 1.5(IQR) = 3.9 + 1.5(1.85) = 6.675 J. Kunto 2.8 10.6 is an outlier. P. Herut 1.1 Graphing b. Use a graphing calculator to make a box- A. Carlos 4.7 Calculator Tip and-whisker plot with and without the C. Franca 3.4 For help with graphing outlier. Which plot represents the data P. Daniels 2.0 box-and-whisker plots, better? S. Roberts 1.1 see Graphing Calculator Lab 4 on page 343. SOLUTION The plot that identifies the outlier is better. There are no values between 4.7 and 10.6. A whisker makes it look like data is distributed throughout that range. Using a plot that identifies the outlier shows that most of the data is very close to Q3. Lesson 54 347  Lesson Practice a. A trainer made a box-and-whisker plot of the number of seconds her (Ex 1) clients could stand on a balance pod. Use the interquartile range to identify outliers. Number of Seconds on Balance Pod 20 30 40 50 60 70 Hint b. Make a box-and-whisker plot to display the data of scores on some state Don’t forget to first put (Ex 2) the numbers in order tests: 411, 507, 387, 475, 507, 477, 484, 605, 496, 504, 529, 585, 459, 586, from least to greatest. 508, 589. Half the tests are between which scores? c. A coach records the number of yards run by his players. (Ex 3) 1, 22, 18, 34, 37, 89, 44, 43, 19, 28, 27, 23, 19, 21 Display the data using a box-and-whisker plot. Identify any outliers. The populations of the 17 largest U.S. cities in 2005 are listed in millions. 0.7, 2, 0.9, 2.8, 1.5, 0.7, 1.3, 0.8, 0.7, 0.6, 0.8, 1.2, 3.8, 1.5, 0.7, 8.1, 0.9 d. Identify any outliers. (Ex 4) e. Use a graphing calculator to make a box-and-whisker plot of this data (Ex 4) with and without the outlier. Which plot represents the data better? Practice Distributed and Integrated Solve each equation. Check your answer. 1. _ 1 +_ 3 x - 5 = 10 _ 1 (26)2 8 2 2. 0.02x - 4 - 0.01x - 2 = -6.3 (24) 3. x - 5x + 4(x - 2) = 3x - 8 (28) Simplify. 4. _ (38) 2x2 - 10x 2x 5. _ (39) d -3 ( b2 _ 3f -3d2 db-2 - _ 4 b-2 ) *6. Chemistry Avogadro’s number is represented by 6.02 × 1023. Writing this number (40) as a product, which value would have to be in the exponent of the expression (6.02 × 1015)(10-)2 ? 7. Multi-Step The formula to convert degrees Fahrenheit to degrees Celsius is (41) C = _59 (F - 32). a. On a separate sheet of paper, make a table of the equivalent Celsius temperature to -4, 32, 50, and 77 degrees Fahrenheit. b. Use the table to make a graph of the relationship. c. Find the slope of the graph. 348 Saxon Algebra 1  *8. A class makes a box-and-whisker plot to show how many children are in each (54) family. Identify the median, upper and lower quartiles, upper and lower extremes, and the interquartile range. Children per Family 0 1 2 3 4 5 *9. A doctor makes a box-and-whisker plot to show the number of patients she (54) sees each day. Identify the median, upper and lower quartiles, upper and lower extremes, and the interquartile range. Patients per Day 10 15 20 25 30 *10. Formulate Create a data set that meets the following criteria: lower extreme 62, (54) lower quartile 70, median 84, upper quartile 86, and upper extreme 95. *11. Multiple Choice Using a box-and-whisker plot, which information can you (54) gather? A the mode B the range C the mean D the number of data values *12. Astronomy The planets’ distances (in millions of miles) from the sun are as follows: (54) 36, 67, 93, 142, 484, 887, 1765, and 2791 Make a box-and-whisker plot of these distances and determine if any planet’s distance is an outlier. 13. Find the percent of increase or decrease to the nearest percent from the original (47) price of $2175.00 to the new price of $2392.50. *14. Choose an appropriate measure of central tendency to represent the data set. Justify (48) your answer. 12 quiz scores (in percents): 86, 92, 88, 100, 86, 94, 92, 78, 90, 96, 94, 84. *15. Manufacturing A skateboard factory has 467 skateboards in stock. The factory can (49) produce 115 skateboards per hour. Write a linear equation in slope-intercept form to represent the number of skateboards in inventory after so many hours if no shipments are made. 16. Write an inequality for the graph below. (50) -9 -7 -5 Lesson 54 349  17. Automotive Maintenance The following chart shows the wear on a particular brand (44) of tires every 10,000 miles. What is the average rate of wear for this brand of tires? Mileage Tread Depth 10,000 20 mm 20,000 16 mm 30,000 12 mm 40,000 8 mm 18. The diagram shows types of transportation. Use the diagram to determine if each (Inv 4) statement is true or false. If the statement is false, provide a counterexample. Transportation Motor Vehicles Trucks Automatic a. If a vehicle is a truck, then the vehicle is an automatic. b. All trucks are motor vehicles. 19. Write Explain the difference between √ -1 and √ 1. (46) cannot be simplified to _6. 2g 1 20. Write Explain why _ 2g + 6 (51) 21. Multiple Choice Which expression is not equivalent to 3rd -1 - _ 6 ? (51) r -1d A -3rd -1 B _-3r C _ -3d r D _ -3 d r -1d *22. Telecommunications Jane bought a prepaid phone card that had 500 minutes. She (52) used about 25 minutes of calling time per week. Write and graph an equation to approximate her remaining calling time y (in minutes) after 9 weeks. 23. Find the slope of the line that passes through (1, 6) and (3,-4). (52) 24. Describe a line that has a slope of 0 and passes through the point (-1, 1). (52) 25. Write an equation in slope-intercept form of a line that passes through the (52) points (14, -3) and (-6, 9). 26. Error Analysis Students were asked to find the sum of the polynomials vertically. (53) Which student is correct? Explain the error. Student A Student B -6x3 - 3x2 + 5 -6x3 - 3x2 + 5 + 2x3 - x - 7 ________ + 2x3 - x - 7 _________ -4x3 - 4x2 - 2 -4x - 3x2 - x - 2 3 350 Saxon Algebra 1  27. Geometry Write a polynomial expression for the perimeter of the triangle. Simplify the (53) polynomial and give your answer in standard form. 2x + 6 4x + 3 3x + 7 3 4 28. Measurement The length of the sidewalk that runs in front of Trina’s house is (53) 3x - 16 and the width is 5x + 21. Find the perimeter of the sidewalk. *29. Multi-Step The table shows the amounts that Doug and Jane plan to deposit in (53) their savings account. Their savings account has the same annual growth rate g. Date 1/1/04 1/1/05 1/1/06 1/1/07 Doug $300 $400 $200 $25 Jane $375 $410 $50 $200 a. On January 1, 2007, the value of Doug’s account D can be modeled by D = 300g3 + 400g2 + 200g + 25, where g is the annual growth rate. Find a model for Jane’s account J on January 1, 2007. b. Find a model for the combined amounts of Doug and Jane’s account on January 1, 2007. 30. Find the sum of (9x3 + 12) + (16x3 - 4x + 2) using a horizontal format. (53) Lesson 54 351  L AB 5 Calculating the Intersection of Two Lines Graphing Calculator Lab (Use with Lesson 55) A graphing calculator can be used to find the intersection of two lines. Find the intersection of y = 2x - 5 and 2y + 3x = 6. Caution 1. Enter the equations into the Y = Editor. Equations entered into the Y = Editor should be solved for y. 2. Graph the equations in a standard viewing Graphing window. Calculator Tip For help with graphing equations, refer to Graphing Calculator Lab 3 on page 305. 3. Approximate the intersection by tracing the line. Trace one of the lines by pressing. The and keys are used to move the cursor to another line. The and keys are used to move the cursor along a line. Use the keys to move the cursor to the intersection. The coordinates of the cursor are displayed at the bottom of the screen. The approximate intersection point of y = 2x - 5 and 2y + 3x = 6 is about (2.3, -0.3). Online Connection www.SaxonMathResources.com 4. Calculate the exact intersection. 352 Saxon Algebra 1  Press and select 5:Intersection. At the prompt “First Curve?,” press to select the first line. At the next prompt, “Second Curve?,” press to select the second line. Use the and keys to move the cursor near the intersection. At the prompt “Guess?,” press. The solution is displayed as a decimal at the bottom of the screen. 5. Change to the x- and y-coordinates to fractions. Press [QUIT] to return to the home screen. Press and , and then press and select 1:>Frac. Press. Press [Y] and , and then press and select 1:>Frac. Press. Lab Practice a. Use the trace feature to find the approximate intersection of y = -2x + 3 and y = 0.5x + 1. b. Use the intersection feature to find the exact intersection of y = -2x + 3 and y = 0.5x + 1. c. Use the graphing calculator to find the intersection of y = -2x + 2 and 3y - 4x = 12. d. Use the intersection feature to find the exact intersection of y = -_32 x - 5 and y = _3 x + 5. 1 e. Use the graphing calculator to find the intersection of y = 4x - 1 and 2y + x = 2. Lab 5 353 LES SON Solving Systems of Linear Equations 55 by Graphing Warm Up 1. Vocabulary The (30) of a linear equation is any ordered pair that makes the equation true. 2. Evaluate 18 + 3n for n = 2. (9) 3. Is (3, 2) a solution to the equation 3x + 2y = 13? Explain. (35) 4. Write 2x + 3y = 6 in slope-intercept form. (49) New Concepts A system of linear equations consists of two or more linear equations containing two or more variables. An example is shown below. Math Language 3x + y = 9 A linear equation is an equation whose graph is x + 2y = 8 a straight line. A solution of a system of linear equations is any ordered pair that makes all the equations true. Example 1 Identifying Solutions Tell whether the ordered pair is a solution of the given system. 3x + 2y = 7 a. (1, 2); x = 7 - 3y SOLUTION Substitute 1 for x and 2 for y in each equation. 3x + 2y = 7 x = 7 - 3y 3(1) + 2(2)  7 1  7 - 3(2) 3+47 17-6 7=7 ✓ 1=1 ✓ The ordered pair (1, 2) makes both equations true. A solution of the linear system is (1, 2). 3x + 2y = 12 b. (2, 3); x = 7 - 3y SOLUTION Substitute 2 for x and 3 for y in each equation. 3x + 2y = 12 x = 7 - 3y 3(2) + 2(3)  12 2  7 - 3(3) 6 + 6  12 27-9 12 = 12 ✓ 2 ≠ -2 ✗ The ordered pair (2, 3) makes only one equation true. Online Connection www.SaxonMathResources.com (2, 3) is not a solution of the system. 354 Saxon Algebra 1 All solutions of a linear equation can be found on its graph. Graphing each equation can solve a system of linear equations. If the system has one solution, the solution is the common point or the point of intersection. Example 2 Solving by Graphing Solve the system by graphing. Then check your solution. y=x+3 y = 2x + 1 SOLUTION Graph both equations on the same coordinate plane. Find the point of intersection. Check that the ordered pair (2, 5) makes both y 6 equations true. (2, 5) 4 y=x+3 y = 2x + 1 y=x+3 2 52+3 5  2(2) + 1 y = 2x + 1 x O 5=5 ✓ 54+1 -2 2 4 5=5 ✓ The ordered pair (2, 5) is a solution of each equation. (2, 5) is a solution of the system. Example 3 Writing Equations in Slope-Intercept Form Solve the system by graphing. Then check your solution. 3x + y = 9 x + 2y = 8 SOLUTION Write the equations in slope-intercept form. 3x + y = 9 x + 2y = 8 -3x __ -3x __ -x ______ -x y = -3x + 9 2y = -x + 8 y = -_ 1x + 4 2 Graph both equations on the same coordinate y 4 plane. Identify the point of intersection. y= _ 1x + 4 (2, 3) 2 2 Check that the ordered pair (2, 3) makes both O x original equations true. -4 -2 2 4 3x + y = 9 x + 2y = 8 -2 y = -3x + 9 3(2) + 3  9 2 + 2(3)  8 -4 6+39 2+68 9=9 ✓ 8=8 ✓ The ordered pair (2, 3) is a solution of each equation. (2, 3) is a solution of the system. Lesson 55 355  Example 4 Solving with a Graphing Calculator Use a graphing calculator to solve the system and then check your solution. 2x + y = -2 y = -3x - 5 SOLUTION Write 2x + y = -2 in slope intercept form. 2x + y = -2 -2x __ = ___ -2x y = -2x - 2 Graphing Graph both equations and then use the Calculator Tip intersection command. For help with graphing The intersection point is (-3, 4). systems, see the Graphing Calculator Check that the ordered pair (-3, 4) makes both Lab 5 on page 352. original equations true. 2x + y = -2 y = -3x - 5 2(-3) + 4  -2 4  -3(-3) - 5 -6 + 4  -2 49-5 -2 = -2 ✓ 4=4 ✓ The ordered pair (-3, 4) is a solution of each equation. (-3, 4) is a solution of the system. Example 5 Application: Rate Plans B