Equations with VARIABLES on BOTH SIDES
Tool Box:
Summary:
Inverse Operations
Properties of Equality
Like Terms
Question:
Equations that contain “variables” on each
side of the equation
To solve these type of equations, first use
the ADDITION or SUBTRACTION
Property of Equality to write an equivalent
equation that has ALL of the variables on
ONE SIDE
Solve an Equation with Variables on
both sides
EXAMPLE: −𝟐 + 𝟏𝟎𝒑 = 𝟖𝒑 − 𝟏
−2 + 10𝑝 = 8𝑝 − 1
1.Write the equation
2. Draw railroad tracks
3. Subtraction Property of Equality
(subtract 8p from both sides of the
equation) 8p is smaller in value than
10p. This allows the variable to remain
positive
4. Simplify/Combine Like Terms
−2 + 10𝑝 = 8𝑝 − 1
- 8p
- 8p
−2 + 2𝑝 =
−1
−2 + 2𝑝 =
+2
−1
+2
𝟐𝒑 =
2𝑝
2
p
5. Isolate the variable
6. Addition Property of Equality
(add +2 to each side of the equation)
7. Simplify/Combine Like Terms
8. Division Property of Equality
(Divide both sides of the equation by 2)
9. Simplify
𝟏
=
1
2
1
=
2
Solve an Equation with Grouping
Symbols
EXAMPLE:
𝟒(𝟐𝒓 − 𝟖) =
4(2𝑟 − 8) =
4(2𝑟 − 8) =
4(2𝑟) − 4(2) =
8𝑟 − 32 =
- 7r
1
7
1
7
1
7
𝟏
𝟕
Use the distribution property to remove
the grouping symbols
(𝟒𝟗𝒓 + 𝟕𝟎)
(49𝑟 + 70)
1. Write the equation
(49𝑟 + 70)
2. Draw railroad tracks
(49𝑟) +
7𝑟 +
- 7r
1
7
(70)
3. Distributive Property
10
4. Subtraction Property of Equality
(subtract 7r from both sides of the
equation)
1r
1r
1r
- 32
- 32
- 32
+ 32
r
=
=
=
10
10
10
+ 32
42
=
NO SOLUTION
5. Combine Like Terms/Simplify
6. Isolate the variable
7. Addition Property of Equality
(add 32 to both sides of the equation)
8. Simplify
Some equations with the variable
on both sides may not have a
solution. That is, there is no value
of the variable that will result in a
true statement
EXAMPLE:
𝟐𝒎 + 𝟓 = 𝟓(𝒎 − 𝟕) − 𝟑𝒎
2𝑚 + 5 = 5(𝑚 − 7) − 3𝑚
2𝑚
2𝑚
2𝑚
2𝑚
- 2m
+
+
+
+
5
5
5
5
=
=
=
=
-
5𝑚
𝟓𝒎
𝟐𝒎
2𝑚
2m
−
−
−
−
35 − 3𝑚
35 − 𝟑𝒎
35
35
5 =
- 35
5 ≠ - 35
Since 5 does not equal - 35, the equation has
NO SOLUTION!!!!
1. Write the equation
2. Draw railroad tracks
3. Distributive Property
4. Combine Like Terms/(5m and -3m)
5. Subtraction Property of Equality
(Subtract 2m from both sides of the
equation)
6. Simplify
7. FALSE STATEMENT 5 ≠ - 35
IDENTITY
Equations do not always have
** The solution of an “identity” is all real exactly one solution. An equation
that is true for all values of the
numbers.
variable is an identity
Example:
𝟑(𝒓 + 𝟏) − 𝟓
3(𝑟 + 1) − 5
= 𝟑𝒓 − 𝟐
= 3𝑟 − 2
𝟑(𝒓) + 𝟑(𝟏) − 𝟓 = 𝟑𝒓 − 𝟐
3𝑟 + 3 − 5
= 3𝑟 − 2
3𝑟 + 3 − 5
𝟑𝒓 − 𝟐
= 3𝑟 − 2
= 𝟑𝒓 − 𝟐
CLEARING THE DENOMINATOR/
FRACTION
1. Write the equation
2. Train Tracks
3. Distributive Property
(Distribute the 3 to each of the terms in
the parentheses)
4. Combine Like Terms ( 3 and – 5)
4. Reflexive Property of Equality
𝟏
EXAMPLE: 𝟒 −
𝟐
𝟑
𝒚 =
𝟑
𝟒
−
𝟏
𝟑
𝒚
𝟏
𝟐
𝟑
𝟏
−
𝒚 =
−
𝒚
𝟒
𝟑
𝟒
𝟑
(𝟏𝟐)
1 Write the equation
1
2
3
1
− (𝟏𝟐) 𝒚 = (𝟏𝟐) − (𝟏𝟐) 𝒚
4
3
4
3
3
3
−
−
+
8𝑦
8𝑦
8y
3
3
- 9
−𝟑
𝟐
=
=
9
9
=
9
=
−
−
+
+
4𝑦
4𝑦
8y
4y
9 + 4y
-9
-6
=
4y
−𝟔
𝟒𝒚
=
𝟒
𝟒
𝟏
𝒐𝒓 − 𝟏 𝟐 = 𝒚
EXAMPLE:
𝒔+ 𝟏
=
𝟖
𝒔
𝟒
𝑠 + 1
𝑠
=
8
4
𝒔 + 𝟏
𝒔
(𝟖)
=
(𝟖)
𝟖
𝟒
𝑠 + 1
𝑠 + 1
- s
1
2. Draw the train tracks
2. Clear the denominator by
multiplying all terms by the common
denominator (12)
3. Simplify
4. Addition Property of Equality
(Add 8y to both sides of the equation)
5. Combine Like Terms/Simplify
6. Isolate the variable
7. Subtraction Property of Equality
(subtract 9 from each side of the
equation)
8. Combine Like Terms/Simplify
9. Division Property of Equality
(Divide both sides of the equation by 4)
10. Simplify
= 2𝑠
= 2𝑠
= - s
=
s
Steps for Solving Equations
Summary
1. Use the Distributive Property to remove the
grouping symbol
2. Simplify the expressions on each side of the
equal sign
3. Use the Addition and/or Subtraction Property of
Equality to get the variables on one side of the
equal sign and the constants (numbers without a
variable) on the other side of the equal sign
1 Write the equation
2. Clear the fraction/Multiply the
TERMS by the common denominator 8
3. Simplify
4. Isolate the variable
5. Subtraction Property of Equality
(subtract s from each side of the
equality)
6. Combine Like Terms/Simplify
4. Simplify the expression on each side of the
equation
5. Use the Multiplication Property or Division
Property to solve.
 If the solution results in a FALSE
STATEMENT, then there is
NO SOLUTION of the equation
 If the solution is an IDENTITY, then the
solution is ALL REAL NUMBERS