2-3 Solving Multi-Step Equations

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Lesson Presentation

California Standards

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2-3 Solving Multi-Step Equations

Warm UpEvaluate 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)

18

–4

11c

0

9m – 14

4x – 5

26 – 4(7 – 5)

2-3 Solving Multi-Step Equations

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.

California Standards

2-3 Solving Multi-Step Equations

Regular price of enrollment

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.

Number ofstudents

Total cost

2-3 Solving Multi-Step Equations

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.

2-3 Solving Multi-Step Equations

Additional Example 1A: Solving Two-Step Equations

Solve the equation. Check your answer.

2x + 1 = 21–1 –1

2x = 20

Since 2x + 1 is divided by 3, multiply both sides by 3 to undo the division.

Since 1 is added to 2x, subtract 1 from both sides to undo the addition.

x = 10

Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.

The solution set is {10}.

2-3 Solving Multi-Step Equations

Additional Example 1A Continued

Solve the equation. Check your answer.

CheckTo check your solution,

substitute 10 for x in the original equation.

7 7

2-3 Solving Multi-Step Equations

Additional Example 1B: Solving Two-Step Equations

Solve the equation. Check your answer.

6 = x

+4 +4 18 = 3x

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.

Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.

The solution set is {6}.

2-3 Solving Multi-Step Equations

Additional Example 1B Continued

Solve the equation. Check your answer.

CheckTo check your solution,

substitute 6 for x in the original equation.

7 7

2-3 Solving Multi-Step EquationsCheck It Out! Example 1a

Solve the equation. Check your answer.

5m + 13 = 2–13 –13

5m = –11

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.

Since m is multiplied by 5, divide both sides by 5 to undo the multiplication.

The solution set is .

2-3 Solving Multi-Step Equations

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

2-3 Solving Multi-Step EquationsCheck It Out! Example 1b

Solve the equation. Check your answer.

–4 –4 –2x = –12

4 – 2x = –8

x = 6

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.

Since x is multiplied by –2, divide both sides by –2 to undo the multiplication.

The solution set is {6}.

2-3 Solving Multi-Step Equations

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

2-3 Solving Multi-Step Equations

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

2-3 Solving Multi-Step Equations

Additional Example 2A: Simplifying Before Solving Equations

Solve 8x – 21 – 5x = –15

8x – 21 – 5x = –15

8x – 5x – 21 = –15

3x – 21 = –15

+21 = +21

3x = 6

x = 2

Use the Commutative Property of Addition. Combine like terms.

Since 21 is subtracted from 3x, add 21 to both sides to undo the subtraction.

Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.

The solution set is {2}.

2-3 Solving Multi-Step EquationsAdditional Example 2B: Simplifying Before Solving

Equations Solve 4 = 2x + 5 – 6x

4 = 2x + 5 – 6x

4 = 2x – 6x + 5

4 = –4x + 5 –5 –5–1 = –4x

Use the Commutative Property of Addition. Combine like terms.

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 .

2-3 Solving Multi-Step Equations

Check It Out! Example 2a

Solve the equation. Check your answer.

2a + 3 – 8a = 8

2a – 8a +3 = 8

–6a + 3 = 8–3 –3

–6a = 5

Use the Commutative Property of Addition. Combine like terms.

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 .

2-3 Solving Multi-Step Equations

Check It Out! Example 2a Continued

Solve the equation. Check your answer.

Check

2a + 3 – 8a = 8

To check your solution,

substitute for a in

the original equation.

8 8

2-3 Solving Multi-Step EquationsCheck It Out! Example 2b

Solve the equation. Check your answer.

–8 – 2d + 2 = 4

–2d –6 = 4 +6 +6

–2d = 10

Use the Commutative Property of Addition. Combine like terms.

Since 6 is subtracted from –2d, add 6 to both sides to undo the subtraction.

–2d + 2 – 8 = 4

d = –5

Since d is multiplied by –2, divide both sides by –2 to undo the multiplication.

The solution set is {–5}.

–8 – 2d + 2 = 4

2-3 Solving Multi-Step Equations

Check It Out! Example 2b Continued

Solve the equation. Check your answer.

Check

To check your solution,

substitute –5 for d in

the original equation.

–8 – 2d + 2 = 4

–8 – 2(–5) + 2 4

–8 + 10 + 2 4

2 + 2 4

4 4

2-3 Solving Multi-Step EquationsCheck It Out! Example 2c

Solve the equation. Check your answer.

6x – 8 = 40 +8 +8

6x = 48

Use the Commutative Property of Addition. Combine like terms.

Since 8 is subtracted from 6x, add 8 to both sides to undo the subtraction.

x = 8

Since x is multiplied by 6, divide both sides by 6 to undo the multiplication.

The solution set is {8}.

4x – 8 + 2x = 40

4x + 2x – 8 = 40

4x – 8 + 2x = 40

2-3 Solving Multi-Step Equations

Check It Out! Example 2c Continued

Solve the equation. Check your answer.

Check

To check your solution,

substitute 8 for x in

the original equation.

32 – 8 + 16 40

24 + 16 40

40 40

4x – 8 + 2x = 40

4(8) – 8 + 2(8) 40

2-3 Solving Multi-Step EquationsAdditional 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

5p – 10 = –15 +10 +10

5p = –5

p = –1

Simplify.

Since 10 is subtracted from 5p, add 10 to both sides.

Since p is multiplied by 5, divide both sides by 5.

The solution set is {–1}.

2-3 Solving Multi-Step Equations

You can think of a negative sign as a coefficient of –1.

–(x + 2) = –1(x + 2) and –x = –1x.

Helpful Hint

2-3 Solving Multi-Step Equations

Additional Example 3B: Simplify Using the Distributive Property

Solve the equation. 10y – (4y + 8) = –20

10y +(–1)(4y + 8) = –20

10y + (–1)(4y) + (–1)(8) = –20

6y – 8 = –20

+8 +8

6y = –12

Distribute –1.

Write subtraction as the addition of the opposite.

Combine like terms.

10y – 4y – 8 = –20 Simplify.

Since 8 is subtracted from 6y, add 8 to both sides to undo the subtraction.

2-3 Solving Multi-Step Equations

Additional Example 3B Continued

Solve the equation. 10y – (4y +8) = –20

6y = –12

y = –2

Since y is multiplied by 6, divide both sides by 6 to undo the multiplication.

2-3 Solving Multi-Step EquationsCheck It Out! Example 3a

Solve the equation. Check your answer.

3(a + 1) – 4 = 5

3(a + 1) – 4 = 5

(3)(a) + (3)(1) – 4 = 5

+ 1 +13a = 6

Distribute 3.

3a + 3 – 4 = 5 Simplify. Combine like terms.

Since 1 is subtracted from 3a, add 1 to both sides to undo the subtraction.

3a – 1 = 5

Since a is multiplied by 3, divide both sides by 3 to undo the multiplication.

a = 2

2-3 Solving Multi-Step Equations

Check It Out! Example 3a Continued

Solve the equation. Check your answer.

Check

To check your solution,

substitute 2 for a in

the original equation.

3(3) – 4 5

9 – 4 5

5 5

3(a + 1) – 4 = 5

3(2 + 1) – 4 5

2-3 Solving Multi-Step EquationsCheck It Out! Example 3b

Solve the equation. Check your answer.

–4(2 – y) = 8

–4(2 – y) = 8

(–4)(2) + (–4)(–y) = 8

+8 +84y = 16

–8 + 4y = 8

Simplify.

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

Distribute –4 .

2-3 Solving Multi-Step Equations

Check It Out! Example 3b Continued

Solve the equation. Check your answer.

–4(2 – y) = 8Check

To check your solution,

substitute 4 for y in

the original equation.

–4(–2) 8

8 8

–4(2 – 4) 8

2-3 Solving Multi-Step EquationsCheck It Out! Example 3c

Solve the equation. Check your answer.

d + 3(d – 4) = 20

Distribute 3.

d + 3(d – 4) = 20

d + 3(d) + 3(–4) = 20

Simplify.d + 3d – 12 = 20

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

4d = 32

d = 8

Since 12 is subtracted from 4d, add 12 to both sides to undo the subtraction.

Since d is multiplied by 4, divide both sides by 4 to undo the multiplication.

2-3 Solving Multi-Step Equations

Check It Out! Example 3c Continued

Solve the equation. Check your answer.

Check

To check your solution,

substitute 8 for d in

the original equation. 20 20

d + 3(d – 4) = 20

8 + 3(8 – 4) 20

8 + 3(4) 20

2-3 Solving Multi-Step EquationsAdditional 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.

G = L – 4

F = 3L

L = L

Since the information is given in relation to Lin, set an equation for each individual in terms of Lin.

2-3 Solving Multi-Step Equations

G + L + F = 51(L – 4) + (L) + (3L) = 51

5L – 4 = 51+4 +4

5L = 55

L = 11

Since 4 is subtracted from 5L add 4 to both sides to undo the subtraction.

Since L is multiplied by 5, divide both sides by 5 to undo the multiplication.

Substitute.

Combine like terms.

Additional Example 4 ContinuedLin 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?

2-3 Solving Multi-Step Equations

Greg sold 7 shirts.

Additional Example 4 Continued

G = L – 4

= 11 – 4

= 7

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?

2-3 Solving Multi-Step Equations

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 = 1319.90119.90)+

12 months

(12m

2-3 Solving Multi-Step Equations

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.

–119.90 –119.9024m = 1200.00

Check It Out! Example 4a Continued

m = 50Sara and Martin each paid $50 per month.

2(12m + 59.95) = 1319.90

2(12m) + 2(59.95) = 1319.90 Distribute 2.

24m + 119.90 = 1319.90

2-3 Solving Multi-Step Equations

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?

number enrolled

is total cost

processing fee

plus

5 = 125.757)+

class cost

(c

Let c represent the cost of the class.

2-3 Solving Multi-Step Equations

Check It Out! Example 4b Continued

5(c + 7) = 125.75

5(c) + 5(7) = 125.75

5c + 35 = 125.75– 35 – 35

5c = 90.75

c = 18.15

The cost per person is $18.15 a month.

Since 35 is added to 5c, subtract 35 from both sides to undo the addition.

Since c is multiplied by 5, divide both sides by 5 to undo the multiplication.

Distribute 5.

2-3 Solving Multi-Step Equations

Lesson Quiz: Part lSolve 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.

19

4

9

25

–28

2-3 Solving Multi-Step Equations

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