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Simplifying and Solving Linear Equations - California Standards Lesson

This lesson covers how to simplify expressions and solve linear equations, including multi-step problems. It is aligned with California Standards grade 4.0 and 5.0.

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Simplifying and Solving Linear Equations - California Standards Lesson

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Evaluate each expression. 1. 2. Simplify each expression. 5. 10c + c 6. 8.2b + 3.8b – 12b 7. 5m + 2(2m – 7) 8. 6x – (2x + 5) –4 26 – 4(7 – 5) 18 11c 0 9m – 14 4x – 5

  3. California Standards 4.0 Students simplify expressions before solvinglinear equations and inequalities in one variable, such as3(2x – 5) + 4(x – 2) = 12. 5.0 Students solve multistep problems, including word problems, involvinglinear equations and linear inequalities in one variable and provide justification for each step.

  4. Regular price of enrollment Number of students Total cost Application fee A martial arts school is offering a special where new students can enroll for half price, after a $12.50 application fee. Ten students enrolled and paid a total of $325. To find the regular price of enrollment, you can solve an equation. 10( +12.50)=325

  5. Notice that this equation contains multiplication, division, and addition. An equation that contains multiple operations will require multiple steps to solve. You will create an equivalent equation at each step.

  6. Additional Example 1A: Solving Two-Step Equations Solve the equation. Check your answer. Since 2x + 1 is divided by 3, multiply both sides by 3 to undo the division. 2x + 1 = 21 Since 1 is added to 2x, subtract 1 from both sides to undo the addition. –1 –1 2x = 20 Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. x = 10 The solution set is {10}.

  7. Additional Example 1A Continued Solve the equation. Check your answer. Check To check your solution, substitute 10 for x in the original equation.  7 7

  8. +4 +4 Additional Example 1B: Solving Two-Step Equations Solve the equation. Check your answer. Since 3x – 4 is divided by 2, multiply both sides by 2 to undo the division. Since 4 is subtracted from 3x, add 4 to both sides to undo the subtraction. 18 = 3x Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 6 = x The solution set is {6}.

  9. Additional Example 1B Continued Solve the equation. Check your answer. Check To check your solution, substitute 6 for x in the original equation.  7 7

  10. –13 –13 The solution set is . Check It Out! Example 1a Solve the equation. Check your answer. Since 5m + 13 is divided by 2, multiply both sides by 2 to undo the division. Since 13 is added to 5m, subtract 13 from both sides to undo the addition. 5m + 13 = 2 5m = –11 Since m is multiplied by 5, divide both sides by 5 to undo the multiplication.

  11. To check your solution, substitute for m in the original equation. Check It Out! Example 1a Continued Solve the equation. Check your answer. Check  1 1

  12. –4 –4 Check It Out! Example 1b Solve the equation. Check your answer. Since 4 – 2x is divided by 4, multiply both sides by 4 to undo the division. Since 4 is added to – 2x, subtract 4 from both sides to undo the addition. 4 – 2x = –8 –2x = –12 Since x is multiplied by –2, divide both sides by –2 to undo the multiplication. x = 6 The solution set is {6}.

  13. Check It Out! Example 1b Continued Solve the equation. Check your answer. Check To check your solution, substitute 6 for x in the original equation. –2 –2 

  14. You may have to combine like terms or use the Distributive Property before you begin solving.

  15. +21 = +21 Additional Example 2A: Simplifying Before Solving Equations Solve 8x – 21 – 5x = –15 8x – 21 – 5x = –15 Use the Commutative Property of Addition. Combine like terms. 8x – 5x – 21 = –15 3x – 21 = –15 Since 21 is subtracted from 3x, add 21 to both sides to undo the subtraction. 3x = 6 Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. x = 2 The solution set is {2}.

  16. –5 –5 –1 = –4x The solution set is . Additional Example 2B: Simplifying Before Solving Equations Solve 4 = 2x + 5 – 6x 4 =2x + 5 – 6x Use the Commutative Property of Addition. Combine like terms. 4 =2x – 6x + 5 4 =–4x+ 5 Since 5 is added to –4x, subtract 5 from both sides to undo the addition. Since x is multiplied by –4, divide both sides by –4 to undo the multiplication.

  17. –3 –3 –6a = 5 Check It Out! Example 2a Solve the equation. Check your answer. 2a + 3 – 8a = 8 Use the Commutative Property of Addition. Combine like terms. 2a – 8a+3 = 8 –6a+ 3 = 8 Since 3 is added to –6a, subtract 3 from both sides to undo the addition. Since a is multiplied by –6, divide both sides by –6 to undo the multiplication. The solution set is .

  18. To check your solution, substitute for a in the original equation. Check It Out! Example 2a Continued Solve the equation. Check your answer. 2a + 3 – 8a = 8 Check  8 8

  19. +6 +6 Check It Out! Example 2b Solve the equation. Check your answer. –8 – 2d + 2 = 4 Use the Commutative Property of Addition. Combine like terms. –8 – 2d+ 2 = 4 –2d+ 2 – 8 =4 –2d–6 = 4 Since 6 is subtracted from –2d, add 6 to both sides to undo the subtraction. –2d = 10 Since d is multiplied by –2, divide both sides by –2 to undo the multiplication. d = –5 The solution set is {–5}.

  20. Check It Out! Example 2b Continued Solve the equation. Check your answer. –8 – 2d + 2 = 4 Check To check your solution, substitute –5 for d in the original equation. –8 – 2(–5) + 2 4 –8 + 10 + 2 4 2 + 2 4  4 4

  21. +8 +8 Check It Out! Example 2c Solve the equation. Check your answer. 4x – 8 + 2x = 40 4x – 8 + 2x = 40 Use the Commutative Property of Addition. Combine like terms. 4x + 2x– 8 = 40 6x– 8 = 40 Since 8 is subtracted from 6x, add 8 to both sides to undo the subtraction. 6x = 48 Since x is multiplied by 6, divide both sides by 6 to undo the multiplication. x = 8 The solution set is {8}.

  22. Check It Out! Example 2c Continued Solve the equation. Check your answer. Check 4x – 8 + 2x = 40 To check your solution, substitute 8 for x in the original equation. 4(8)– 8 + 2(8) 40 32 – 8 + 16 40 24 + 16 40  40 40

  23. +10 +10 Additional Example 3A: Simplify Using the Distributive Property Solve the equation. 5(p – 2) = –15 5(p – 2) = –15 Distribute 5. 5(p) + 5(–2) = –15 Simplify. 5p – 10 = –15 Since 10 is subtracted from 5p, add 10 to both sides. 5p = –5 Since p is multiplied by 5, divide both sides by 5. p = –1 The solution set is {–1}.

  24. Helpful Hint You can think of a negative sign as a coefficient of –1. –(x + 2) = –1(x + 2) and –x = –1x.

  25. +8 +8 Additional Example 3B: Simplify Using the Distributive Property Solve the equation. 10y – (4y + 8) = –20 Write subtraction as the addition of the opposite. 10y +(–1)(4y + 8) = –20 Distribute –1. 10y + (–1)(4y) + (–1)(8) = –20 10y– 4y– 8 = –20 Simplify. 6y – 8 = –20 Combine like terms. Since 8 is subtracted from 6y, add 8 to both sides to undo the subtraction. 6y = –12

  26. Additional Example 3B Continued Solve the equation. 10y – (4y +8) = –20 6y = –12 Since y is multiplied by 6, divide both sides by 6 to undo the multiplication. y = –2

  27. + 1 +1 Check It Out! Example 3a Solve the equation. Check your answer. 3(a + 1) – 4 = 5 3(a + 1) – 4 = 5 Distribute 3. (3)(a) + (3)(1) – 4 = 5 3a+ 3 – 4= 5 Simplify. Combine like terms. 3a– 1 = 5 Since 1 is subtracted from 3a, add 1 to both sides to undo the subtraction. 3a = 6 Since a is multiplied by 3, divide both sides by 3 to undo the multiplication. a = 2

  28. Check It Out! Example 3a Continued Solve the equation. Check your answer. 3(a + 1) – 4 = 5 Check To check your solution, substitute 2 for a in the original equation. 3(2 + 1) – 4 5 3(3) – 4 5 9 – 4 5  5 5

  29. +8 +8 4y = 16 Check It Out! Example 3b Solve the equation. Check your answer. –4(2 – y) = 8 –4(2 – y) = 8 Distribute –4 . (–4)(2) + (–4)(–y) = 8 Simplify. –8 +4y= 8 Since –8 is added to 4y, add 8 to both sides. Since y is multiplied by 4, divide both sides by 4 to undo the multiplication. y = 4

  30. Check It Out! Example 3b Continued Solve the equation. Check your answer. –4(2 – y) = 8 Check To check your solution, substitute 4 for y in the original equation. –4(2 – 4) 8 –4(–2) 8 8 8 

  31. +12 +12 Check It Out! Example 3c Solve the equation. Check your answer. d + 3(d – 4) = 20 d + 3(d – 4) = 20 d + 3(d) + 3(–4) = 20 Distribute 3. Simplify. d + 3d – 12 = 20 Combine like terms. 4d – 12 = 20 Since 12 is subtracted from 4d, add 12 to both sides to undo the subtraction. 4d = 32 Since d is multiplied by 4, divide both sides by 4 to undo the multiplication. d = 8

  32. Check It Out! Example 3c Continued Solve the equation. Check your answer. Check d + 3(d – 4) = 20 To check your solution, substitute 8 for d in the original equation. 8 + 3(8 – 4) 20 8 + 3(4) 20 20 20 

  33. Additional Example 4: Application Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell? To determine the number of shirts sold write an equation: G + L + F = 51. Since the information is given in relation to Lin, set an equation for each individual in terms of Lin. G = L – 4 F = 3L L = L

  34. +4 +4 Additional Example 4 Continued Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell? G + L + F = 51 (L – 4) + (L) + (3L) = 51 Substitute. 5L – 4 = 51 Combine like terms. Since 4 is subtracted from 5L add 4 to both sides to undo the subtraction. 5L = 55 Since L is multiplied by 5, divide both sides by 5 to undo the multiplication. L = 11

  35. Additional Example 4 Continued Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell? G = L – 4 = 11 – 4 = 7 Greg sold 7 shirts.

  36. initial fee for 2 12 months Monthly fee for 2 is total cost. plus + (12m 119.90) 2 = 1319.90 Check It Out! Example 4a At a local gym, there is a joining fee of $59.95 and a monthly membership fee. Sara and Martin both joined this gym. Their combined cost for 12 months was $1319.90. How much is the monthly fee? Let m represent the monthly fee paid by each.

  37. –119.90 –119.90 24m = 1200.00 Check It Out! Example 4a Continued 2(12m + 59.95) = 1319.90 Distribute 2. 2(12m) + 2(59.95) = 1319.90 24m + 119.90 = 1319.90 Since 119.90 is added to 24m, subtract 119.90 from both sides to undo the addition. Since m is multiplied by 24, divide both sides by 24 to undo the multiplication. m = 50 Sara and Martin each paid $50 per month.

  38. class cost processing fee number enrolled is total cost plus + (c 7) 5 = 125.75 Check It Out! Example 4b Lily and 4 of her friends want to enroll in a yoga class. After enrollment, the studio requires a $7 processing fee. The 5 girls pay a total of $125.75. How much does the class cost? Let c represent the cost of the class.

  39. – 35 – 35 5c = 90.75 Check It Out! Example 4b Continued 5(c + 7) = 125.75 5(c) + 5(7) = 125.75 Distribute 5. Since 35 is added to 5c, subtract 35 from both sides to undo the addition. 5c + 35 = 125.75 Since c is multiplied by 5, divide both sides by 5 to undo the multiplication. c = 18.15 The cost per person is $18.15 a month.

  40. Lesson Quiz: Part l Solve each equation. 1. 2y + 29 – 8y = 5 2. 3(x – 9) = 30 3.x – (12 – x) = 38 4. 5. If 3b – (6 – b) = –22, find the value of 7b. 4 19 25 9 –28

  41. Lesson Quiz: Part ll 6. Josie bought 4 cases of sports drinks for an upcoming meet. After talking to her coach, she bought 3 more cases and spent an additional $6.95 on other items. Her receipts totaled $74.15. Write and solve an equation to find how much each case of sports drinks cost. 4c + 3c + 6.95 = 74.15; $9.60

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