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ALGEBRA 1 LESSON 2-3
(For help, go to Lessons 1-2 and 1-7.)
Solving Multi-Step Equations
Simplify each expression.
1. 2n – 3n
2. –4 + 3b + 2 + 5b
3. 9(w – 5)
4. –10(b – 12)
5. 3(–x + 4)
6. 5(6 – w)
Evaluate each expression.
7. 28 – a + 4a for a = 5
8. 8 + x – 7x for x = –3
9. (8n + 1)3 for n = –2
10.–(17 + 3y) for y = 6
2-3
ALGEBRA 1 LESSON 2-3
Solving Multi-Step Equations
Solutions
1. 2n – 3n = (2 – 3)n = –1n = –n
2. –4 + 3b + 2 + 5b = (3 + 5)b + (–4 + 2) = 8b – 2
3. 9(w – 5) = 9w – 9(5) = 9w – 45
4. –10(b – 12) = –10b – (–10)(12) = –10b + 120
5. 3(–x + 4) = 3(–x) + 3(4) = –3x + 12
6. 5(6 – w) = 5(6) – 5w = 30 – 5w
7. 28 – a + 4a for a = 5: 28 – 5 + 4(5) = 28 – 5 + 20 = 23 + 20 = 43
8. 8 + x – 7x for x = –3: 8 + (–3) – 7(–3) = 8 + (–3) + 21 = 5 + 21 = 26
9. (8n + 1)3 for n = –2: (8(–2) + 1)3 = (–16 + 1)3 = (–15)3 = –45
10. –(17 + 3y) for y = 6: –(17 + 3(6)) = –(17 + 18) = –35
2-3
ALGEBRA 1 LESSON 2-3
Solving
Multi-Step
Equations
Solve 3a + 6 + a = 90
4a + 6 = 90
Combine like terms.
4a + 6 – 6 = 90 – 6
Subtract 6 from each side.
4a = 84
Simplify.
4a
84
=
4
4
Divide each side by 4.
a = 21
Simplify.
Check: 3a + 6 + a = 90
3(21) + 6 + 21
90
63 + 6 + 21
90
Substitute 21 for a.
90 = 90
2-3
ALGEBRA 1 LESSON 2-3
Solving
Multi-Step
Equations
You need to build a rectangular pen in your back yard for
your dog. One side of the pen will be against the house. Two sides of
the pen have a length of x ft and the width will be 25 ft. What is the
greatest length the pen can be if you have 63 ft of fencing?
Relate: length
of side
plus 25 ft plus
length equals
of side
amount
of fencing
Define: Let x = length of a side adjacent to the house.
Write: x + 25 + x = 63
2-3
ALGEBRA 1 LESSON 2-3
Solving Multi-Step Equations
(continued)
x + 25 + x = 63
2x + 25 = 63
Combine like terms.
2x + 25 – 25 = 63 – 25
Subtract 25 from each side.
2x = 38
Simplify.
2x
38
=
2
2
Divide each side by 2.
x = 19
The pen can be 19 ft long.
2-3
ALGEBRA 1 LESSON 2-3
Solving
Multi-Step
Equations
Solve 2(x – 3) = 8
2x – 6 = 8
Use the Distributive Property.
2x – 6 + 6 = 8 + 6
Add 6 to each side.
2x = 14
Simplify.
2x
14
=
2
2
Divide each side by 2.
x = 7
Simplify.
2-3
ALGEBRA 1 LESSON 2-3
Solving
Multi-Step
Equations
Solve
+
= 17
3x
2
x
5
Method 1: Finding common denominators
3x x
2 + 5 = 17
3
1
x
+
2
5 x = 17
15
2
x
+
10
10 x = 17
Rewrite the equation.
3
1
A common denominator of 2 and 5
is 10.
17
10 x = 17
10 17
10
(
x
)
=
17 10
17 (17)
Combine like terms.
Multiply each each by the reciprocal
of 17 , which is 10 .
10
x = 10
Simplify.
2-3
17
ALGEBRA 1 LESSON 2-3
Solving
Multi-Step
Equations
Solve
+
= 17
3x
2
x
5
Method 2: Multiplying to clear fractions
3x x
2 + 5 = 17
3x x
10( 2 + 5 ) = 10(17)
3x
Multiply each side by 10, a common
multiple of 2 and 5.
x
10( 2 ) + 10( 5 ) = 10(17)
Use the Distributive Property.
15x + 2x = 170
Multiply.
17x = 170
Combine like terms.
17x
170
=
17
17
Divide each side by 17.
x = 10
Simplify.
2-3
ALGEBRA 1 LESSON 2-3
Solving
Multi-Step
Equations
Solve 0.6a + 18.65 = 22.85.
100(0.6a + 18.65) = 100(22.85)
100(0.6a) + 100(18.65) = 100(22.85)
60a + 1865 = 2285
60a + 1865 – 1865 = 2285 – 1865
60a = 420
60a
420
=
60
60
a = 7
The greatest of decimal
places is two places.
Multiply each side by 100.
Use the Distributive
Property.
Simplify.
Subtract 1865 from each side.
Simplify.
Divide each side by 60.
Simplify.
2-3
ALGEBRA 1 LESSON 2-3
Solving Multi-Step Equations
Solve each equation.
1. 4a + 3 – a = 24 7
2. –3(x – 5) = 66 –17
3. n + n = 7 12
3
4
4. 0.05x + 24.65 = 27.5 57
2-3
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
Sides
Simplify.
(For help, go to Lessons 1-7 and 2-3.)
1. 6x – 2x
2. 2x – 6x
3. 5x – 5x
4. –5x + 5x
Solve each equation.
5. 4x + 3 = –5
6. –x + 7 = 12
7. 2t – 8t + 1 = 43
8. 0 = –7n + 4 – 5n
2-4
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
Solutions
Sides
1.
6x – 2x = (6 – 2)x = 4x
2.
3.
5x – 5x = (5 – 5)x = 0x = 0
4. –5x + 5x = (–5 + 5)x = 0x = 0
5.
4x + 3 = –5
4x = –8
x = –2
7. 2t – 8t + 1 =
–6t + 1 =
–6t =
t=
6.
43
43
42
–7
8.
2-4
2x – 6x = (2 – 6)x = –4x
–x + 7 = 12
–x = 5
x = –5
0
0
12n
n
=
=
=
=
–7n + 4 – 5n
–12n + 4
4
1
3
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
The measure ofSides
an angle is (5x – 3)°. Its vertical angle has
a measure of (2x + 12)°. Find the value of x.
5x – 3 = 2x + 12
Vertical angles are congruent.
5x – 3 – 2x = 2x + 12 – 2x
Subtract 2x from each side.
3x – 3 = 12
Combine like terms.
3x – 3 + 3 = 12 + 3
Add 3 to each side.
3x = 15
Simplify.
3x = 15
3
3
Divide each side by 3.
x = 5
Simplify.
2-4
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
You can buy a skateboard
for $60 from a friend and rent the
Sides
safety equipment for $1.50 per hour. Or you can rent all items you
need for $5.50 per hour. How many hours must you use a
skateboard to justify buying your friend’s skateboard?
Relate: cost of plus equipment equals skateboard and equipment
friend’s
rental
rental
skateboard
Define: let h = the number of hours you must skateboard
Write:
60
+
1.5 h
=
2-4
5.5 h
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
Sides
(continued)
60 + 1.5h = 5.5h
60 + 1.5h – 1.5h = 5.5h – 1.5h
Subtract 1.5h from each side.
60 = 4h
Combine like terms.
60
4h
=
4
4
Divide each side by 4.
15 = h
Simplify.
You must use your skateboard for more than 15 hours to justify buying the
skateboard.
2-4
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
Solve each equation.
Sides
–6z + 8 = z + 10 – 7z
–6z + 8 = z + 10 – 7z
–6z + 8 = –6z + 10
Combine like terms.
–6z + 8 + 6z = –6z + 10 + 6z Add 6z to each side.
8 = 10
Not true for any value of z!
This equation has no solution
b.
4 – 4y = –2(2y – 2)
4 – 4y = –2(2y – 2)
4 – 4y = –4y + 4
Use the Distributive Property.
4 – 4y + 4y = –4y + 4 + 4y
Add 4y to each side.
4 = 4
Always true!
The equation is true for every value of y, so the equation is an identity.
a.
2-4
ALGEBRA 1 LESSON 2-4
Equations with Variables on Both
Sides
Solve each equation.
1. 3 – 2t = 7t + 4 – 1
9
2. 4n = 2(n + 1) + 3(n – 1) 1
3. 3(1 – 2x) = 4 – 6x no solution
4. You work for a delivery service. With Plan A, you can earn
$5 per hour plus $.75 per delivery. With Plan B, you can earn
$7 per hour plus $.25 per delivery. How many deliveries must
you make per hour with Plan A to earn as much as with Plan
B? 4 deliveries
2-4