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Linear Equations

Linear Equations

- Solving linear equations
- Whatever you do to one side of an equation, you

must also do to the other. - Goal is to get the variable alone on one side of

the equation by undoing the listed operations.

Linear Equations

- Examples
- X 4 8
- 4 4 subtract 4 (or add 4) to

both sides - X 4
- To check substitute your solution for the

variable and see if the statement is true. - X 4 8, X 4 therefore
- 4 4 8 (true)

Linear Equations

- 4x 48
- 4 4 since 4 is multiplying x we

divide both - sides by 4 to make the

coefficient of x 1 - x 12
- Simplify first to get only one term with the

variable! - 2x 3x 13 2
- 5x 15 simplify by combining

like terms - 5 5 divide both sides by 5
- x 3

Linear Equations

- Two operations
- 2x 4 8 nothing to simplify
- 4 4 subtract 4 from both sides
- 2x 4
- 2 2 divide both sides by 2
- x 2

Linear Equations

- Variables on both sides of equation
- 9 4x 8x cant simplify since 4x and 8x are
- on opposite sides of the equation. Remember

the goal is to get one term with the variable! - 9 4x 8x
- 4x 4x add 4x to both sides to

get the x - terms together
- 9 12x
- 12 12 divide both sides by 12
- ¾ x

Linear Equations

- Variables on both sides (2 methods)
- 9x 2 6x 10 Method 1 move constants first
- 2 2 Subtract 2 from both

sides - 9x 6x 12
- 6x 6x subtract 6x from both sides
- 3x 12
- 3 3 divide both sides by 3
- x 4

Linear Equations

- Variables on both sides (2 methods)
- 9x 2 6x 10 Method 2 move variables

first - 6x 6x subtract 6x from both

sides - 3x 2 10
- 2 2 subtract 2 from both

sides - 3x 12
- 3 3 divide both sides

by 3 - x 4

Linear Equations

- Applying the Distributive Property
- 28 7x 3(x 4)
- 28 7x ( 3)x ( 4) change subtract to

add opp. - 28 7x ( 3x) 12 distribute the 3 to

all terms - 28 4x 12 simplify
- 12 12 subtract 12

from both sides - 16 4x
- 4 4 divide both

sides by 4 - 4 x

Linear Equations

- Applying the Distributive Property
- 6 ( x 3 ) 2x think of the before the

parenthesis as a 1 - 6 1(x 3) 2x
- 6 1x 3 2x distribute the 1 thru the

parenthesis - 3 1x 2x simplify
- 1x 1x add 1x to both sides
- 3 3x
- 3 3 divide both sides by 3
- 1 x

Linear Equations

- Special cases (1)
- 4x 3 3x 3 x
- 4x 3 4x 3 simplify
- 4x 4x subtract 4x from both sides
- 3 3
- Since 3 DOES equal 3, we say that ALL real

numbers are solutions!

Linear Equations

- Special cases (2)
- 3x 7 x 2x 11
- 3x 7 3x 11 simplify
- 3x 3x subtract 3x from both sides
- 7 11
- Since 7 DOES NOT equal 11, we say that there

is NO solution to this equation. In other words,

no real number will make this equation true!

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