CONFIDENTIAL 1 Solving Two-Step and Multi-Step Equations.

CONFIDENTIAL 1

Solving Two-Step andSolving Two-Step andMulti-Step EquationsMulti-Step Equations

CONFIDENTIAL 2

Warm UpWarm Up

Simplify:

1) K = 8 42) z = -9 33) -2 = w

-74) 6 = t

-5

1) k = 32

2) z = -27

3) w = 14

4) t = - 30

CONFIDENTIAL 3

IntroductionEquations containing more than one operation can model real-world situations, such as the cost of a

music club membership.

Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card

allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40.

To find the number of CDs c that Alex bought, you can solve an equation.

$3.95c + $19.95 = $63.40

Cost of discount card

Cost per CD Total cost

Next page

CONFIDENTIAL 4

Notice that this equation contains multiplication and addition. Equations that contain more than one operation

require more than one step to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse

operations and work backward to undo them one at a time.

$3.95c + $19.95 = $63.40

Operations in the Equation To Solve

1) First c is multiplied by 3.95.

2) Then 19.95 is added.

1) Subtract 19.95 from both sides of the equation.

2) Then divide both sides by 3.95.

CONFIDENTIAL 5

Solving Two-Step EquationsSolve 10 = 6 - 2x. Check your answer.

10 = 6 - 2x First x is multiplied by -2. Then 6 is added.- 6 -6 Work backward: Subtract 6 from both sides.

4 = -2x 4 = -2x-2 -2-2 = x

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

Check 10 = 6 - 2x101010

6 - 2 (-2)6 - (-4)

10

CONFIDENTIAL 6

Solve each equation. Check your answer.Now you try!

1) -4 + 7x = 3

2) 1.5 = 1.2 y - 5.7

3) n + 2 = 2 7 1) x = 1

2) y = 6

3) n = 0

CONFIDENTIAL 7

Solving Two-Step Equations That Contain Fractions

Solve: q – 1 = 3 15 5 5

Method 1: Use fraction operations.

q – 1 = 315 5 5

Since 1/5 is subtracted from q/5 15 , add 1/5 to both sides to undo the subtraction.– 1 - 1

5 5 q = 4 15 5

15 × q = 15 × 4 = 60 15 5 5

Since q is divided by 15, multiply both sides by 15 to undo the division.

Next page

Simplify.q = 12

CONFIDENTIAL 8

Method 2: Multiply by the least common denominator (LCD) to clear the fractions.

q – 1 = 315 5 5

Multiply both sides by 15, the LCD of the fractions. q – 1 = 3

15 5 515 × 15 ×

q – 1 = 315 5 5

15 × 15 × 15 × Distribute 15 on the left side.

q - 3 = 9 + 3 + 3

q = 12

Simplify

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

CONFIDENTIAL 9

Solve each equation. Check your answer.

Now you try!

1) 2x – 1 = 5 5 2

2) 3u + 1 = 7 4 2 8

3) n - 1 = 8 5 3 3

3) n = 15

1) x = 554

2) u = 12

CONFIDENTIAL 10

Simplifying Before Solving Equations

Solve each equation.

A) 6x + 3 - 8x = 136x + 3 - 8x = 13 Use the Commutative

Property of Addition.

Since 3 is added to -2x, subtract 3 from both sides to undo the addition.

Combine like terms.6x - 8x + 3 = 13

- 2x + 3 = 13 - 3 - 3

-2x = 10Since x is multiplied by -2, divide both sides by -2 to undo the multiplication.

-2x = 10 -2 -2

x = -5

CONFIDENTIAL 11

B) 9 = 6 - (x + 2)

9 = 6 - 2 - x

5 = -x-1 -1

Write subtraction as addition of the opposite.

-5 = x

Since 4 is added to -x, subtract 4 from both sides to undo the addition.

9 = 6 + (-1) (x + 2)9 = 6 + (-1)(x) + (-1)(2)9 = 6 - x - 2

9 = 4 - x

5 = -x -4 -4

Use the Commutative Property of Addition. Combine like terms.

Since x is multiplied by -1, divide both sides by -1 to undo the multiplication.

Simplify

Distribute -1 on the right side.

CONFIDENTIAL 12

Now you try!

1) 2a + 3 - 8a = 8

2) -2 (3 - d) = 4

Solve each equation. Check your answer.

3) 4 (x - 2) + 2x = 403) x = 8

1) x = -56

2) d = 5

CONFIDENTIAL 13

Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card allows

them to buy CDs for $3.95 each. After one year, Alex has spent $63.40. Write and solve an equation to find

how many CDs Alex bought during the year.

Let c represent the number of CDs that Alex purchased.That means Alex has spent $3.95c.

However, Alex must also add the amount spent on the card.

total cost = cost of compact discs + cost of discount cardi.e., 63.40 = 3.95c + 19.95

Make a Plan

CONFIDENTIAL 14

Solve63.40 = 3.95c + 19.95

-19.95 -19.9543.45 = 3.95c43.45 = 3.95c 3.95 3.95

11 = cAlex bought 11 CDs during the year.

The cost per CD is about $4, so if Alex bought 11 CDs, this amount is about 11(4) = $44.

Add the cost of the discount card, which is about $20: 44 + 20 = 64. So the total cost was about $64, which is

close to the amount given in the problem, $63.40.

Check

CONFIDENTIAL 15

Sara paid $15.95 to become a member at a gym. She then paid a monthly membership fee. Her total

cost for 12 months was $735.95. How much was the monthly fee?

Now you try!

$ 60

CONFIDENTIAL 16

Solving Equations to Find an Indicated Value

If 3a + 12 = 30, find the value of a + 4.

Since 12 is added to 3a, subtract 12 from both sides to undo the addition.

Step1: Find the value of a.3a + 12 = 30

-12 -12 3a = 18

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

3a = 183 3

Step2: Find the value of a + 4.a + 46 + 4

10

To find the value of a + 4, substitute 6 for a.

Simplify.

CONFIDENTIAL 17

Now you try!

1) If 2x + 4 = -24, find the value of 3x.

Solve each equation. Check your answer.

1) x = -42

CONFIDENTIAL 18

Assignments

1) 4a + 3 = 11

2) 8 = 3r - 1

3) 42 = -2d + 6

4) x + 0.3 = 3.3

Solve each equation. Check your answer:

1) a = 2

2) r = 3

3) d = -16

4) x = 3

CONFIDENTIAL 19

5) 15y + 31 = 61

6) 9 - c = -13

7) x + 4= 15 6

8) y + 1 = 5 3 4 12

9) If 3x - 13 = 8, find the value of x - 4.

Solve each equation. Check your answer:

5) y = 2

6) c = 22

7) x = 66

8) y = 1

9) x - 4 = 5

CONFIDENTIAL 20

10) Paul bought a student discount card for the bus. The card cost $7 and allows him to buy daily bus

passes for $1.50. After one month, Paul spent $29.50.How many daily bus passes did Paul buy?

10) 15

CONFIDENTIAL 21

Equations containing more than one operation can model real-world situations, such as the cost of a

music club membership.

Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card

allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40.

To find the number of CDs c that Alex bought, you can solve an equation.

$3.95c + $19.95 = $63.40

Cost of discount card

Cost per CD Total cost

Next page

Let’s review

CONFIDENTIAL 22

Notice that this equation contains multiplication and addition. Equations that contain more than one operation

require more than one step to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse

operations and work backward to undo them one at a time.

$3.95c + $19.95 = $63.40

Operations in the Equation To Solve

1) First c is multiplied by 3.95.

2) Then 19.95 is added.

1) Subtract 19.95 from both sides of the equation.

2) Then divide both sides by 3.95.

review

CONFIDENTIAL 23

Solving Two-Step EquationsSolve 10 = 6 - 2x. Check your answer.

10 = 6 - 2x First x is multiplied by -2. Then 6 is added.- 6 -6 Work backward: Subtract 6 from both sides.

4 = -2x 4 = -2x-2 -2-2 = x

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

Check 10 = 6 - 2x101010

6 - 2 (-2)6 - (-4)

10

review

CONFIDENTIAL 24

Solving Two-Step Equations That Contain Fractions

Solve: q – 1 = 3 15 5 5

Method 1: Use fraction operations.

q – 1 = 315 5 5

Since 1/5 is subtracted from q/5 15 , add 1/5 to both sides to undo the subtraction.– 1 - 1

5 5 q = 4 15 5

15 × q = 15 × 4 = 60 15 5 5

Since q is divided by 15, multiply both sides by 15 to undo the division.

Next page

Simplify.q = 12

review

CONFIDENTIAL 25

Method 2: Multiply by the least common denominator (LCD) to clear the fractions.

q – 1 = 315 5 5

Multiply both sides by 15, the LCD of the fractions. q – 1 = 3

15 5 515 × 15 ×

q – 1 = 315 5 5

15 × 15 × 15 × Distribute 15 on the left side.

q - 3 = 9 + 3 + 3

q = 12

Simplify

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

review

CONFIDENTIAL 26

Simplifying Before Solving Equations

Solve each equation.

A) 6x + 3 - 8x = 136x + 3 - 8x = 13 Use the Commutative

Property of Addition.

Since 3 is added to -2x, subtract 3 from both sides to undo the addition.

Combine like terms.6x - 8x + 3 = 13

- 2x + 3 = 13 - 3 - 3

-2x = 10Since x is multiplied by -2, divide both sides by -2 to undo the multiplication.

-2x = 10 -2 -2

x = -5

review

CONFIDENTIAL 27

B) 9 = 6 - (x + 2)

9 = 6 - 2 - x

5 = -x-1 -1

Write subtraction as addition of the opposite.

-5 = x

Since 4 is added to -x, subtract 4 from both sides to undo the addition.

9 = 6 + (-1) (x + 2)9 = 6 + (-1)(x) + (-1)(2)9 = 6 - x - 2

9 = 4 - x

5 = -x -4 -4

Use the Commutative Property of Addition. Combine like terms.

Since x is multiplied by -1, divide both sides by -1 to undo the multiplication.

Simplify

Distribute -1 on the right side.

review

CONFIDENTIAL 28

Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card allows

them to buy CDs for $3.95 each. After one year, Alex has spent $63.40. Write and solve an equation to find

how many CDs Alex bought during the year.

Let c represent the number of CDs that Alex purchased.That means Alex has spent $3.95c.

However, Alex must also add the amount spent on the card.

total cost = cost of compact discs + cost of discount cardi.e., 63.40 = 3.95c + 19.95

Make a Plan

review

CONFIDENTIAL 29

Solve63.40 = 3.95c + 19.95

-19.95 -19.9543.45 = 3.95c43.45 = 3.95c 3.95 3.95

11 = cAlex bought 11 CDs during the year.

The cost per CD is about $4, so if Alex bought 11 CDs, this amount is about 11(4) = $44.

Add the cost of the discount card, which is about $20: 44 + 20 = 64. So the total cost was about $64, which is

close to the amount given in the problem, $63.40.

Check

review

CONFIDENTIAL 30

Solving Equations to Find an Indicated Value

If 3a + 12 = 30, find the value of a + 4.

Since 12 is added to 3a, subtract 12 from both sides to undo the addition.

Step1: Find the value of a.3a + 12 = 30

-12 -12 3a = 18

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

3a = 183 3

Step2: Find the value of a + 4.a + 46 + 4

10

To find the value of a + 4, substitute 6 for a.

Simplify.

review

CONFIDENTIAL 31

You did a great job You did a great job today!today!