LINEAR EQUATIONS

Definitions

Like terms are terms that have the same
variables with the same exponents

2x, 7x
− 5x², 8x²

Unlike terms have different variables or different
exponents on the variables .

2x, 7
− 5x³ , 6x²

Properties of the Real Numbers

Combine Like Terms

8x + 4x = (8 + 4)x = 12x

5y− 6y = (5 − 6)y = −y

x + 15 − 5x + 9 = (1− 5)x + (15+9) = −4x + 24

1 3x + 2 + 6y − 4 + 7x = (3 + 7)x + 6y + (2 − 4) = 10x + 6y − 2

Addition and Multiplication Properties of
Equality

Addition Property of Equality
For all real numbers A, B, and C, the equations
A = B and A + C = B + C
are equivalent .
That is, the same number may be added to each side of an
equation without changing the solution set.

Multiplication Property of Equality
For all real numbers A, and B, and for C ≠ 0, the equations
A = B and AC = BC
are equivalent.
That is, each side of the equation may be multiplied by the
same nonzero number without changing the solution set .

Solving Equations

	Use addition property to “move” 9 to the other side
	“undo” subtraction with addition

Check: x− 9 = 24
33 − 9 = 24 ?
24 = 24 true

Find the solution to the equation

	Use the addition Property to “move” 12 to other side.
	“undo” addition with subtraction

Check: x+ 12 = 31
19 + 12 = 31 ?
31 = 31 true

Multiplication Property of Equality
For all real numbers A, and B, and for C ≠ 0, the equations
A = B and AC = BC
are equivalent.
That is, each side of the equation may be multiplied by the same nonzero number without
changing the solution set.

Find the solution to the equation

Use Multiplication Property
“Undo” division with multiplication

Division Property of Equality
For all real numbers A, and B, and for C ≠ 0, the equations
A = B and
are equivalent.
That is, each side of the equation may be divided by the same nonzero number without
changing the solution set.

Find the solution to the equation

4x = 48 “Undo” multiplication with division

Strategies for Solving Linear Equations

1. Eliminate fractions : multiply both sides by LCD or
Multiply LCD by each term.

2. Remove ( ) or grouping symbols using the Distributive
property.

3. Combine Like terms.

4. Move variables to same side.

5. Isolate the variable.

Solve 3x− 4 = 17

	“Undo” subtraction with addition
	“Undo” multiplication with division

Solve 21 = 6 + 3(x + 2)

	1. Remove () using the distributive property.
	2. Combine like terms
	3. “Undo” addition with subtraction.

Solve 8x + 3 = 6x + 21

	1. Move variables and constants to the same side.
	2. Undo multiplication with division

Solve 6(x− 2) + 2x + 3 = 4(2x − 3) + 2

	1. Remove () with distributive Property.
	2. Combine like terms.
	3. Move variables to one side and constants to the other.
	NOTICE: The variable disappeared!

‐9 does not equal ‐10!!!

A FALSE statement means the equation has no solution.
Therefore, equation is inconsistent.

Solve 4(x + 1) − 6(x + 2) = −2(x + 4)

	1. Remove () with distributive Property.
	2. Combine like terms.
	3. Move variables to one side and constants to the other.
	NOTICE: The variable disappeared!

Since the statement is true,
0 = 0 or ‐8 = ‐8, the solution is all real
numbers.