Solving Linear Equations

Solve the following nonstandard linear equation for x and check your answer by substitution.

x - 3 = 4x -18
x - 3 + 3 = 4x - 18 + 3	Add 3 to both sides
x = 4x -15	Collect like terms
x - 4x = 4x - 4x -15	Subtract 4x from both sides
-3x = -15	Collect like terms
	Divide both sides by 3
x = 5

CHECK: Substitute 5 for each x in the original equation

and the solution checks

*** MY RULE THAT MAKES IT EASIER TO SOLVE THESE EQUATIONS!! ***

YOU CAN MOVE A TERM FROM ONE SIDE TO THE OTHER AS LONG AS YOU CHANGE ITS SIGN !!!! THE OBJECT IS TO GET ALL OF THE x TERMS ON THE LEFT SIDE OF THE EQUATION AND ALL OF THE CONSTANT TERMS ON THE RIGHT SIDE OF THE EQUATION.

EXAMPLE: Solve 2x + 6 = 3x + 5

2x - 3x = +5 - 6

Note:

I left 2x on the left side -> sign stays same

I moved 3x from the right side to the left side -> its sign changed

I left +5 on the right side -> sign stays same

I moved +6 from the left side to the right side -> its sign changed

Now all the x terms are on the left and all the constant terms are on the right. So collect like terms and solve.

-x = -1 but you need to solve for x not -x!!

So multiply both sides of the equation through by -1. Thus

(-1)(-x) = (-1)(-1) x = 1 is your answer !!!

CHECK: Substitute 1 for each x in the original equation

2x + 6 = 3x + 5 Original equation

Substitute 1 for x

and thus your solution checks !!!

If an equation involves fractions, get rid of the fractions first by multiplying both sides of the equation through by their lowest common denominator - the smallest number that all the denominators will divide into evenly.

Solve: Multiply each term in the equation by 6 which is the LCD of 6 and 3

x + 2 = 6

x = 6 - 2

x = 4

CHECK: Original equation

Substitute 4 for x

1 = 1 so the solution checks !!!!!

If an equation involves decimals, get rid of the decimals first by multiplying both sides of the equation through by an appropriate power of 10 - that is, use 10, 100, 1000, etc

EXAMPLE: .3x + 1.5 = 8.4

If we multiply this equation by 10, it won't have any more decimals!

.3x + 1.5 = 8.4	Original equation
10(.3x + 1.5) = 10(8.4)	Multiply both sides by 10
3x + 15 = 84
3x = 84 - 15	Move terms to correct side
3x = 69	Collect terms
	Divide both sides by 3
x = 23

Check:.3x + 1.5 = 8.4 Original equation

Substitute 23 for x

8.4 = 8.4 and the solution checks !

ONE FRACTION is set equal to ANOTHER FRACTION

If one single fraction is set equal to another single fraction, then you can simply cross multiply! That is, multiply one numerator by the opposite denominator and set that equal to the other numerator multiplied by the other denominator. Then solve as a normal equation!

EXAMPLES:

Note: one fraction is set equal to another 14 7 fraction

7(x+4) = 2(14)

7x + 28 = 28

7x = 28 - 28

7x = 0

x = 0

Note: One fraction is set equal to another fraction

9(2x - 8) = 4(x + 3)

18x -72 = 4x + 12

18x - 4x = 72 + 12

14x = 84

x = 6