EdGems Course 2 - Lesson 4.2 - Solving Two-Step Equations Student Lesson PDF

Lesson 4.2 Solving Two-Step Equations I can solve two-step equations. Deborah opened an ice cream stand at the local pool last summer. She spent $148 on a freezer to start her business. She earned a profit of $0.75 for each ice cream cone sold. She earned a total of $452 during the summer. Let x rep...

Lesson 4.2 Solving Two-Step Equations I can solve two-step equations. Deborah opened an ice cream stand at the local pool last summer. She spent $148 on a freezer to start her business. She earned a profit of $0.75 for each ice cream cone sold. She earned a total of $452 during the summer. Let x represent the number of cones Deborah sold. The equation that models this situation is: 0.75x − 148 = 452 This equation is called a two-step equation. Two-step equations have two different operations in the equation. The two-step equation above has both multiplication and subtraction. In order to get the variable by itself in a two-step equation, you must perform two inverse operations. The inverse operations must undo the order of operations. That means you must start by undoing any addition or subtraction first and then use inverse operations to remove any multiplication or division. E xa m p l e 1 Solution Using the equation at the beginning of this lesson, determine the number of ice cream cones, x, Deborah sold last summer. 0.75x − 148 = 452 Write the equation. Add 148 to both sides. Divide each side by 0.75. 0.75x − 148 = 452 +148 +148 0.75x 600 = 0.75 0.75 x = 800 Check the solution. ☑ 0.75(800) − 148 =? 452 600 − 148 =? 452 452 = 452 Deborah sold 800 ice cream cones last summer. 64 Lesson 4.2 ~ Solving Two-Step Equations FL_EdGems_Math_Course_2.indb 64 4/19/21 2:56 PM E xa m p l e 2 Solution x + 2 = −6 7 Solve the equation for x. Check the solution. Write the equation. Subtract 2 from both sides of the equation. Multiply each side by 7. x + 2 = −6 7 −2 −2 x = −56 −56 + 2 =? −6 ☑ Check the solution. 7 −8 + 2 =? −6 −6 = −6 x = −56 E xa m p l e 3 Solution x = −8 ∙ 7 7 7∙ 1 John had 10 2 cups of flour left in his flour bag. He made some batches of 3 3 brownies which each used cup of flour. There were 3 cups left in his flour 4 4 bag. How many batches of brownies did he make? Write an equation for the situation. Rewrite the equation by adding the opposite. 1 Subtract 10 2 from both sides. Multiply both sides by the reciprocal 3 of − 4. 1 10 2 − 3 b 4 3 = 34 10 2 + (− 4 b) = 3 4 1 3 3 1 1 −10 2 −10 2 ( )( ) ( )( ) 4 −3 3 3 − 4 b = −6 4 4 −3 b=9 John made 9 batches of brownies. Exercises Solve each equation for the variable. Check each solution. h +1=−4 2 1. 7x + 3 = 24 2. 4. −3m + 6 = 30 5. 6.4y + 1 = −31 3. x −7=3 5 6. 0.25x + 10 = 12 Lesson 4.2 ~ Solving Two-Step Equations FL_EdGems_Math_Course_2.indb 65 65 4/19/21 2:56 PM Solve each equation for the variable. Check each solution. 7. 14 = p 4 1 + 8 2 8. 1 p 2 − 3 = 9 9. w +8=3 −4 10. a − 1.5 = 0.75 2 11. 6 − 3 w = 46 12. 30x − 14 = 116 13. 9 8 = −x + 10 5 14. −3 = 13 − 2d 15. 4 f 4 + 3 4 1 = 14 16. Min solved three problems incorrectly. For each problem, describe the error he made and determine the correct answers. a. −4x + 1 = 21 −1 −1 −4x = 20 4 4 x=5 b. 45 = 20 − 5x +20 + 20 65 = −5x −5 −5 c. x = −13 x 3 − 2 = 22 +2 +2 3 ÷ _​  3x​ = 24 ÷ 3 x=8 Write an equation for each statement and then solve each equation. 17. Twice a number x, then increased by nine, is twenty-five. Find the number. 18. Four more than the quotient of the number y and negative three is eleven. Find the number. 19. Twelve decreased by the product of five and a number w is 52. Find the number. 20. Three less than one-half of a number p is 4. Find the number. 21. Jeannie got a new pair of eyeglasses. She was able to pay $15 at the time she received her glasses. She will pay $12 each month on the balance. The original cost of the glasses was $87. a. Let x represent the number of months needed to pay off the cost of the glasses. Write an equation and explain how the equation represents this situation. b. How many months will it take Jeannie to pay for her glasses? Explain how you know your answer is correct. 22. Each month the fire department hosts a pancake feed. This month 75 people attended the pancake feed. This was 13 less than twice as many people as were at the pancake feed last month. How many people attended last month’s pancake feed? Write and use an equation to find the answer. 23. Michael owns his own business selling hand-crafted birdhouses. He sells each birdhouse for $22. He charges $8 for shipping no matter how many birdhouses a person orders. A customer ordered birdhouses with a total bill of $206, including shipping. Write and solve an equation to determine how many birdhouses this customer ordered. 66 Lesson 4.2 ~ Solving Two-Step Equations FL_EdGems_Math_Course_2.indb 66 4/19/21 2:56 PM w 24. The perimeter of a rectangle is found using the formula P = 2l + 2w where P is the perimeter, l is the length and w is the width. A rectangle has a perimeter of 70 centimeters and a side length of 12 centimeters. What is the width of the rectangle? l 25. Ned had $178. He purchased six video games that each cost the same amount. He then had $67 left after the purchase. Let c be the cost of one video game. Write an equation and solve to find the value of c. 26. Solve the equation x + 2y = 10 for y. 27. Write a real-world problem that can be represented by the equation 2x + 50 = 140. 28. Hampton said 2x 5 = 12 was a one-step equation because he could solve it in one step. Tiffany said it was a two-step equation because she could solve it using two operations. Explain how each of them are correct. Review 29. Simplify each expression. a. −6(x − 1) b. 2 + 10x + 1 − 3x c. 3(1 − 4x) d. 5 + 7x − 9 e. −4x − 2 + 8x f. −9(2x + 8) 30. An oil drilling rig started drilling at ground level at 8 AM. It ran constantly and was at −75 feet at 6 PM. If it continued to run constantly, at what time will the rig reach oil if the oil is located at −135 feet? 31. A root beer bottle factory finds it acceptable to have 1.5% of their products test as defective. Yesterday, they found they produced 28 defective bottles out of the 2,000 produced. Are they under the acceptable limit? Explain how you know. Lesson 4.2 ~ Solving Two-Step Equations FL_EdGems_Math_Course_2.indb 67 67 4/19/21 2:56 PM