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NAME _____________________________________________ DATE ____________________________ PERIOD ____________ Lesson 1 Reteach Solve Equations with Rational Coefficients To solve an equation when the coefficient is a rational number, multiply each side by the multiplicative inverse of the fraction. Example 𝟒 Solve 𝟕 𝒙 = 𝟏𝟔. Check your solution. 4 𝑥 7 7 4 4 7 • 1 4 𝑥 7 1 = 16 Write the equation. 7 4 ( ) • 𝑥 = ( ) • 16 1 7 4 1 7 =4 1 • 4 16 1 𝑥 = 28 4 𝑥 7 Check = 16 4 (28) ≟ 16 7 4 4 28 ( ) ≟ 16 7 1 1 16 = 16 Multiply each side by the multiplicative inverse of Write 16 𝑎𝑠 16 1 4 7 , 7 4 . Divide out common factors. Simplify. Write the original equation. Replace x with 28. Write 28 as 28 1 . Divide out common factors. This sentence is true. Solve each equation. Check your solution. 1. 1 𝑥 6 3 =4 4. 4 𝑤 = 1 6 21 30 7. − 𝑥 = −5 5 6 2. 𝑛 = 15 3 5. 5 𝑡 = 12 9 4 8. 𝑟 = Course 3 • Chapter 2 Equations in One Variable 27 32 2 3 14 15 1 1 3 3. 𝑑 = 6. 8 𝑎 = 2 5 9. − 𝑚 = 4 23 NAME _____________________________________________ DATE ____________________________ PERIOD ____________ Lesson 2 Reteach Solve Two-Step Equations A two-step equation contains two operations. To solve a two-step equation, undo each operation in reverse order. Example 1 Solve 2a + 6 = 14. Check your solution. 2a + 6 = 14 – 6 = –6 2a = 2𝑎 2 Check 8 8 =2 Write the equation. Subtraction Property of Equality Simplify. Division Property of Equality a=4 Simplify. 2a + 6 = 14 Write the equation. 2(4) + 6 ≟ 14 14 = 14 Replace a with 4 to see if the sentence is true. The sentence is true. The solution is 4. Sometimes it is necessary to combine like terms before solving an equation. Example 2 Solve 5 = 8x – 2x – 7. Check your solution. 5 = 6x – 7 5 + 7 = 6x – 7 + 7 Write the equation. Addition Property of Equality 12 = 6x Simplify. 12 6 Division Property of Equality = 6𝑥 6 2=x Simplify. The solution is 2. Check this solution. Exercises Solve each equation. Check your solution. 1. 2d + 7 = 9 2. 11 = 3z + 5 3. 2s – 4 = 6 4. –12 = 5r + 8 5. – 6p – 3 = 9 6. –14 = 4x – 2 7. 2c + 2 = 10 8. 3 + 9n = 21 9. 21 = 5 – r 10. 8 – 5b = –7 11. –10 = 6 – 4m 12. –3t + 4 = 19 𝑎 13. 2 + 6 = 5 Course 3 • Chapter 2 Equations in One Variable 1 14. − 3 𝑞 − 7 = −3 15. 4 − 𝑣 5 =0 23 NAME _____________________________________________ DATE ____________________________ PERIOD ____________ Lesson 3 Reteach Write Two-Step Equations Some verbal sentences translate into two step equations. Example 1 Translate each sentence into an equation. Sentence Equation Four more than three times a number is 19. 3𝑛 + 4 = 19 Five is seven less than twice a number. 5 = 2𝑛 – 7 Seven more than the quotient of a number and 3 is10. 7 + 3 = 10 𝑛 After a sentence has been translated into a two step equation, you can solve the equation. Example 2 Translate the sentence into an equation. Then find the number. Thirteen more than five times a number is 28. Words Thirteen more than five times a number is 28. Variable Let n = the number. Equation 5n + 13 = 28 –13 = –13 Write the equation. Subtraction Property of Equality 5n = 15 Simplify. 5𝑛 5 Division Property of Equality = 15 5 n=3 Therefore, the number is 3. Exercises Define a variable. Then translate each sentence into an equation. Then find each number. 1. Five more than twice a number is 7. 2. Fourteen more than three times a number is 2. 3. Seven less than twice a number is 5. 4. Two more than four times a number is –10. 5. Eight less than three times a number is –14. 6. Three more than the quotient of a number and 2 is 7. Course 3 • Chapter 2 Equations in One Variable 23 NAME _____________________________________________ DATE ____________________________ PERIOD ____________ Lesson 4 Reteach Solve Equations with Variables on Each Side Some equations, like 3x – 9 = 6x, have variables on each side of the equals sign. Use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation. Example 1 Solve 3x – 9 = 6x. Check your solution. 3x – 9 = 6x 3x – 3x – 9 = 6x – 3x Write the equation. Subtraction Property of Equality –9 = 3x Simplify by combining like terms. –3 = x Mentally divide each side by 3. To check your solution, replace x with –3 in the original equation. 3x – 9 = 6x Write the equation. 3(–3) – 9 ≟ 6(–3) Replace x with –3. Check –18 = –18 The sentence is true. The solution is –3. Example 2 Solve 4a – 7 = 5 – 2a. 4a – 7 = 5 – 2a 4a + 2a – 7 = 5 – 2a + 2a 6a – 7 = 5 6a – 7 + 7 = 5 + 7 6a = 12 a=2 The solution is 2. Write the equation. Addition Property of Equality Simplify by combining like terms. Addition Property of Equality Simplify. Mentally divide each side by 6. Check this solution. Exercises Solve each equation. Check your solution. 1. 6s – 10 = s 2. 8r = 4r – 16 3. 25 – 3u = 2u 4. 14t – 8 = 6t 5. k + 20 = 9k – 4 6. 11m + 13 = m + 23 7. –4b – 5 = 3b + 9 8. 6y – 1 = 27 – y 9. 1.6h – 72 = 4h – 30 10. 8.5 – 3z = –8z 11. 10x + 8 = 5x – 3 12. 16 – 7d = –3d + 2 Course 3 • Chapter 2 Equations in One Variable 23 NAME _____________________________________________ DATE ____________________________ PERIOD ____________ Lesson 5 Reteach Solve Multi-Step Equations Example 1 Solve 2(4a – 5) = 30. 2(4a – 5) = 30 Write the equation. 8a – 10 = 30 Distributive Property 8a – 10 + 10 = 30 + 10 Addition Property of Equality 8a = 40 Simplify. 8𝑎 8 Division Property of Equality = 40 8 a=5 Simplify. Example 2 BOOKS Roland has 3 paperback books and 4 hardcover books. Each hardcover book is worth $11 more than each paperback book. If the value of all of his books is $79, what is the cost of one paperback book? Write an equation to represent the bar model. 3p + 4(p + 11) = 79 Write the equation. 3p + 4p + 44 = 79 Distributive Property 7p + 44 = 79 Simplify. 7p + 44 + (–44) = 79 + (–44) Addition Property of Equality 7p = 35 7𝑝 7 = Simplify. 35 7 Division Property of Equality p=5 Simplify. So, the cost of one paperback book is $5. Exercises Solve each equation. Check your solution. 1. 2(3b – 1) = 40 2. 49 = –7(t + 1) 3. 5(1 – n) = 75 4. 4(x – 2) = 3(x – 3) 5. –5(p + 2) = 2(2p – 15) + p 6. 4z – 6 = 6(z + 2) + 8 Course 3 • Chapter 2 Equations in One Variable 23