Preview Warm Up California Standards Lesson Presentation

Warm Up Evaluate each expression. 1. 2. –4 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 9m – 14 4x – 5

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

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. Regular price of enrollment Number of students 10( +12.50)=325 Total cost Application fee

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.

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}.

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

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. +4 +4 18 = 3x Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 6 = x The solution set is {6}.

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 

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 –13 –13 5m = –11 Since m is multiplied by 5, divide both sides by 5 to undo the multiplication. The solution set is .

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

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 –4 –4 –2x = –12 Since x is multiplied by –2, divide both sides by –2 to undo the multiplication. x = 6 The solution set is {6}.

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 

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

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. +21 = +21 3x = 6 Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. x = 2 The solution set is {2}.

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 –5 –5 –1 = –4x 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. The solution set is .

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. –3 –3 –6a = 5 Since a is multiplied by –6, divide both sides by –6 to undo the multiplication. The solution set is .

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

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. +6 +6 –2d = 10 Since d is multiplied by –2, divide both sides by –2 to undo the multiplication. d = –5 The solution set is {–5}.

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

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. +8 +8 6x = 48 Since x is multiplied by 6, divide both sides by 6 to undo the multiplication. x = 8 The solution set is {8}.

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

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. +10 +10 5p = –5 Since p is multiplied by 5, divide both sides by 5. p = –1 The solution set is {–1}.

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

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 10y + (–1)(4y) + (–1)(8) = –20 Distribute –1. 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. +8 +8 6y = –12

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

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. + 1 +1 3a = 6 Since a is multiplied by 3, divide both sides by 3 to undo the multiplication. a = 2

Check It Out! Example 3a Continued Solve the equation. Check your answer. Check 3(a + 1) – 4 = 5 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

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. +8 +8 4y = 16 Since y is multiplied by 4, divide both sides by 4 to undo the multiplication. y = 4

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

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. d + 3d – 12 = 20 Simplify. 4d – 12 = 20 Combine like terms. +12 +12 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

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

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

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. +4 +4 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

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.

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. Monthly fee for 2 is total cost. initial fee for 2 plus 2 = 1319.90 119.90) + 12 months (12m

Check It Out! Example 4a Continued Distribute 2. 24m + 119.90 = 1319.90 –119.90 –119.90 24m = 1200.00 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.

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. number enrolled is total cost processing fee plus 5 = 125.75 7) + class cost (c

Check It Out! Example 4b Continued Distribute 5. 5c + 35 = 125.75 Since 35 is added to 5c, subtract 35 from both sides to undo the addition. – 35 – 35 5c = 90.75 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.

5. If 3b – (6 – b) = –22, find the value of 7b. 4 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

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