Solving Equations Notes
Goal: To find a solution to the equation by isolating the variable (or getting the variable alone) on one
side of the equation .
Recall:
• Equations are mathematical statements that use an equal sign to show that two expressions have the
same value.
• Equations are solved, by finding the value of the variable.
• The value of the variable is called the solution.
• Adding and subtracting are inverse operations.
• Multiplying and dividing are inverse operations.
• Squaring and taking the square root are inverse operations.
Notes:
1. Work is shown vertically, or down the paper.
2. The equal signs should be lined up.
3. Use properties to justify your work. (The Equality Property is often used.)
4. Division is shown with a fraction bar.
5. Multiplication is shown to the sides on the equations, not below them. Do NOT use an x to represent
multiplication.
6. Use reverse order of operations.
7. It is often easier to simplify each side (by combining like terms and/or distributing) before using the
equality property to simplify.
Example One:
5 + x = 12
−5
−5
!
0+ x = 7
x=7
Example Two:
2x
= −10
3
3 2x
3
= −10 ⋅
! ⋅
2 3
2
1x = −15
x = −15
Given
Equality Property of Subtraction
Inverse Property of Addition *
Identity Property of Addition
Given
Equality Property of Multiplication
Inverse Property of Multiplication *
Identity Property of Multiplication
−2 + 3x = 16
+2
+2
0 + 3x = 18
Example Three: !
3x = 18
3x 18
=
3
3
1x = 6
x=6
6x − 1 = −19
+1 +1
6x + 0 = −18
Example Four: !
6x = −18
6x −18
=
6
6
1x = −3
x = −3
Example Five:
Solving Equations Notes
Given
Equality Property of Addition
Inverse Property of Addition *
Identity Property of Addition
Equality Property of Division
Inverse Property of Multiplication *
Identity Property of Multiplication
Given
Equality Property of Addition
Inverse Property of Addition *
Identity Property of Addition
Equality Property of Division
Inverse Property of Multiplication *
Identity Property of Multiplication
6x + 2x − 8 + 1 = 10x − 5 − 6
8x − 7 = 10x − 11
− 8x
− 8x
0 − 7 = 2x − 11
− 7 = 2x − 11
!
+ 11
+ 11
4 = 2x + 0
4 = 2x
4 2x
=
2 2
2 = 1x
2=x
Given
Combine like terms
Equality Property of Subtraction
Inverse Property of Addition *
Identity Property of Addition
Equality Property of Addition
Inverse Property of Addition *
Identity Property of Addition
Equality Property of Division
Inverse Property of Multiplication *
Identity Property of Multiplication
*You will be skipping this step (marked with an asterisk) in the future.