Multi-Step Equations
Tool Box:
Summary:
Inverse Operations
Properties of Equality
Like Terms
Question:
Multi-Step Equations contain “more than
one” operation
To solve multi-step equations, Simplify
one or both sides of the equation by
combining like terms, then use inverse
operations to isolate the variable
Solve by “Combining Like Terms”
𝟑𝒕 + 𝟓𝒕 − 𝟓
EXAMPLE:
3𝑡 + 5𝑡 − 5 =
8𝑡
8𝑡
8t
−5
−5
+5
8𝑡
11
=
=
11
11
+5
=
16
1.Write the equation
2. Draw railroad tracks
3 Combine Like Terms 3t and 5t
3. Isolate the variable
4. Addition Property of Equality
(add 5 to both sides of the equation)
5. Simplify/Combine Like Terms
6. Division Property of Equality
(Divide both sides of the equation by 8)
7. Simplify
16
=
8
= 𝟏𝟏
8
Solve by using the Distributive Prop
𝟓𝒂
EXAMPLE:
+ 𝟑(𝒂 + 𝟐) = 𝟐𝟐
𝟓𝒂
+ 𝟑(𝒂 + 𝟐) = 𝟐𝟐
5𝑎
+ 3(𝑎 + 2) = 22
5a
+
8a
3a + 6
+ 6
= 22
= 22
8a
+ 6
- 6
= 22
= - 6
8a
= 16
8𝑎
=
8
a
=
16
8
2
1. Write the equation
2. Draw railroad tracks
3. Distributive Property (Distribute the
3 to each of the terms in the
parentheses)
4 Combine Like Terms (5a and 3a)
5. Isolate the variable
6. Subtraction Property of Equality
(subtract 6 from both sides of the
equation)
7. Simplify/Combine Like Terms
8. Division Property of Equality
(Divide both sides of the equation by 8)
9. Simplify
Solve by Multiplying by a
Reciprocal
(multiplicative inverse)
𝟑
EXAMPLE:
𝟑
𝟒
𝟒
(𝒂 − 𝟓) = 𝟗
(𝒂 − 𝟓) = 𝟗
1 Write the equation
𝟒 𝟑
2 Draw railroad tracks
3. Multiplication Property of Equality
4
Multiply each side of the equation by 3
𝟒
(𝟑) 𝟒 (𝒂 − 𝟓) = 𝟗 (𝟑)
3
the reciprocal of 4
3. Simplify
4. Addition Property of Equality
(Add 5 to both sides of the equation)
5. Combine Like Terms/Simplify
a − 5 = 12
a − 5 = 12
+ 5
+5
a
= 17
Equations with Negative Coefficients
4
Example:
𝟒 −
𝒙
−
=
𝟏𝟎
𝟒 − 𝟏𝒙 =
𝟒 + (−𝟏𝒙) =
-4
𝟏𝟎
𝟏𝟎
- 4
𝑥
=
10
1. Write the equation
2. Train Tracks
3. Identity Prop. x = -1x
4. Subtraction Property
5. Subtraction Property of Equality
(Subtract 4 from both sides of the
equation
6. Simplify/Combine Like Terms
7. Division Property of Equality
(Divide both side of the equation by –1)
8. Simplify
- 1x =
6
−𝟏𝒙
𝟔
=
−𝟏
−𝟏
x
= -6
Combine Like Terms Before Solving
Example:
𝑚 − 5𝑚 + 3 = 47
𝑚 − 5𝑚 + 3 = 47
𝟏𝑚 − 5𝑚 + 3 = 47
− 𝟒𝒎 + 𝟑 = 𝟒𝟕
- 3
- 3
- 4m
= 44
−𝟒𝒎
=
−𝟒
m
=
𝟒𝟒
−𝟒
-11
1. Write the equation
2. Train Tracks
3. Identity Property m = 1m
4. Combine Like Terms 1m and -5m
5. Subtraction Property of Equality
(Subtract 3 from each side of the
equation)
6. Simplify/Combine Like Terms
7. Division Property of Equality
(Divide both sides of the equation
by – 4)
8. Simplify
Try It!
1.
2.
37 = 4d + 5
8y + 6 – 9y = -4
𝑘
3.
5
− 10 = 3
4.
40 = 2(10 + 4𝑘) + 2𝑘
5.
5
6
(2𝑦 + 4) = 10
FUNCTIONS
y is a function of x or y depends on x
y = 3x
- 15
Dependent variable
Independent Variable
Range
Domain
Output
Input
EXAMPLE: The output of a function is 2 more
than 4 times the input. Find the input when the
output is 14.
Output is 2 more than
“y”
= “2 + “
14
=
2 +
Solve the equation
14 = 4x +
14
-2
=
4x
12
12
4
3
=
4x
4𝑥
4
x
=
=
+
-
4x
2
2
2
4 times input
“4x”
Write an equation for the function. Let
y be the output and x be the input
“Is” means equal to, “more than”
means to add, “times” means to
multiply
Substitute “14” in for the output “y”
1. Write the equation
2. Draw the railroad track
3. Isolate the variable
4. Subtraction Property of Equality
(subtract 2 from each side of the
equation
5. Combine Like Terms/Simplify
6. Division Property of Equality
(divide both sides of the equation by 4)
7. Simplify
TRY IT!!
The output of a function is 3 less than 6 times the
input. Find the input when the output is 15