12-3 Solving Equations with Variables on Both Sides

Warm UpSolve.

1. 6n + 8 – 4n = 202. –4w + 16 – 4w = –323. 25t – 17 – 13t = 674. 12 = 2(x + 7) + 4

n = 6w = 6t = 7x = –3

12-3 Solving Equations with Variables on Both Sides

Problem of the DayYou buy 1 cookie on the first day, 2 on the second day, 3 on the third day, and so on for 10 days. Your friend pays $10 for a cookie discount card and then buys 10 cookies at half price. You both pay the same total amount. What is the cost of one cookie? $0.20

12-3 Solving Equations with Variables on Both Sides

Learn to solve equations that have variables on both sides.

12-3 Solving Equations with Variables on Both Sides

Group the terms with variables on one side of the equal sign, and simplify.

Example 1:

A. 60 – 4y = 8y

60 – 4y + 4y = 8y + 4y60 = 12y

Add 4y to both sides. (If you subtracted 8y From both sides, you Would have 0 left on The right and you doNot want that.)

Simplify.

60 – 4y = 8y

12 12

5 = y

12-3 Solving Equations with Variables on Both Sides

Solve.

Example 2

7c = 2c + 557c = 2c + 55

7c – 2c = 2c – 2c + 555c = 555c = 555 5c = 11

Subtract 2c from both sides.Simplify.

Divide both sides by 5.

12-3 Solving Equations with Variables on Both Sides

Additional Example 2B

Solve.49 – 3m = 4m + 1449 – 3m = 4m + 14

49 – 3m + 3m = 4m + 3m + 1449 = 7m + 14

49 – 14 = 7m + 14 – 1435 = 7m35 = 7m7 75 = m

Add 3m to both sides.Simplify.Subtract 14 fromboth sides.

Divide both sides by 7.

12-3 Solving Equations with Variables on Both Sides

Additional Example 2C

25 x = 1

5 x – 12

25 x = 1

5 x – 1225 x 1

5

– x = 1 5 x – 121

5 x–

15 x –12=15 x (5)(–12)=(5)

x = –60

Subtract 15 x from both

sides.Simplify.

Multiply both sides by 5.

12-3 Solving Equations with Variables on Both Sides

Solve.8f = 3f + 658f = 3f + 65

8f – 3f = 3f – 3f + 655f = 655f = 655 5f = 13

Subtract 3f from both sides.Simplify.

Divide both sides by 5.

Check It Out: Example 2A

12-3 Solving Equations with Variables on Both Sides

Solve.54 – 3q = 6q + 9

54– 3q = 6q + 954 – 3q + 3q = 6q + 3q + 9

54 = 9q + 954 – 9 = 9q + 9 – 9

45 = 9q

5 = q

Add 3q to both sides.Simplify.Subtract 9 from both sides.

Divide both sides by 9.

Check It Out: Example 2B

45 =9 9

9q

12-3 Solving Equations with Variables on Both Sides

23 w = 1

3 w – 9

23w = 1

3 w – 923 w 1

3

– w = 1 3 w – 91

3 w–

13 w –9=13 w (3)(–9)=(3)

w = –27

Subtract 13w from both

sides.

Simplify.

Multiply both sides by 3.

Check It Out: Example 2CSolve.

12-3 Solving Equations with Variables on Both Sides

Christine can buy a new snowboard for $136.50. She will still need to rent boots for $8.50 a day. She can rent a snowboard and boots for $18.25 a day. How many days would Christine need to rent both the snowboard and the boots to pay as much as she would if she buys the snowboard and rents only the boots for the season?

Additional Example 3: Consumer Math Application

12-3 Solving Equations with Variables on Both Sides

Additional Example 3 Continued

18.25d = 136.5 + 8.5d18.25d – 8.5d = 136.5 + 8.5d – 8.5d

9.75d = 136.59.75d = 136.59.75 9.75

d = 14

Let d represent the number of days.

Subtract 8.5dfrom both sides.Simplify.Divide both sides by 9.75.

Christine would need to rent both the snowboard and the boots for 14 days to pay as much as she would have if she had bought the snowboard and rented only the boots.

12-3 Solving Equations with Variables on Both Sides

1. Group the terms with variables on one side of the equal sign, and simplify. 19p = 10p + 54.

A. 9p = –54B. 29p = –54C. 29p = 54D. 9p = 54

Lesson Quiz for Student Response Systems

12-3 Solving Equations with Variables on Both Sides

2. Solve.73 + 4m = –7m – 26

A. m = –9B. m = –7C. m = 9D. m = 7

Lesson Quiz for Student Response Systems

12-3 Solving Equations with Variables on Both Sides

3. Matthew orders T-shirts from an online store for $25 each and an additional $20 for shipping. Mike buys the same kind of T-shirts at a local shop for $27 each. If Matthew and Mike spent the same amount, how many T-shirts did each of them buy?

A. 5 T-shirtsB. 10 T-shirtsC. 15 T-shirtsD. 20 T-shirts

Lesson Quiz for Student Response Systems