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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

a(b + c) = ab + ac Distributive property
a + b = b + a Commutative property of
addition
ab = ba Commutative property of
multiplication
(a + b) + c = a + (b + c) Associative property of
addition
(ab)c = a(bc) Associative prop

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.

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