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CONFIDENTIAL 1
Solving Two-Step andSolving Two-Step andMulti-Step EquationsMulti-Step Equations
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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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.
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CONFIDENTIAL 17
Now you try!
1) If 2x + 4 = -24, find the value of 3x.
Solve each equation. Check your answer.
1) x = -42
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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CONFIDENTIAL 31
You did a great job You did a great job today!today!