In solving equations there are many steps to go through and rules to follow. Generally in solving equations, get the term with the variable by itself and then solve for the variable. Ask the question “Is the variable by itself?”. If the answer is no, “What is with it and what is it doing with it?”. After asking, and answering these questions, “undo” or solve the equation. Use the following rules.

 

Addition undoes Subtraction and vice versa

Multiplication undoes division and vice versa

 

More advanced rules will follow later.

 

Remember- What is done to one side of the equation (everything on either the left or right of the equal sign), MUST be done to the other!

 

Example

 

x + 5 = 7

 

This problem can be understood as “what number(x) plus 5 equals 7?”. Ask “Is x by itself?”. No, it has a 5 with it (on that side of the equation).”What is it doing with it (the 5)?”. It is adding since it is + 5. Next, ask what undoes addition. It is subtraction. So subtract 5 from both sides.

x + 5 = 7

 

Subtract 5 from both sides

 

x = 2

 

“Is x by itself?”

 

Yes, so the answer is x = 2.

 

 

 

Example

x – 5 = 10

 

Add five to both sides.

x = 15

 

x is by itself so the answer is x = 15.

 

 

 

Example

 

4x = 12

 

To undo the multiplication with 4, divide by 4.

 

=

 

On the left side, 4 divided by 4 is 1, and 1 times anything (here it is x) is itself (x), so the 4’s cancel out on the left side.

 

=3.

 

x = 3

 

Answer!

 

 

 

 

Example

 

= 12

 

To undo the division of 6, multiply by 6.

 

(6)= 12 (6)

 

The 6’s on the left side cancel out and 12 (6) is 72.

 

x = 72

 

Answer!

 

 

 

Now do some multi-step problems. Always get the term with the variable, x in this case, by itself first.

 

 

Example

 

-3x – 4 = 5

 

Undo the –4 first. So add 4 to both sides. This will get the x term by itself.

-3x = 9

 

Undo the multiplication by dividing by -3.

=

The -3’s on the left cancel and = -3.

 

x = -3

 

Answer!

 

 

 

 

 

Example

+ 7 = 11

 

Subtract 7 from both sides.

= 4

 

Multiply both sides by 3, since x is being divided by 3.

 

(3) = 4(3)

 

The 3’s on the left cancel and 4(3) = 12.

 

x = 12

 

Answer!

 

 

 

 

Sometimes things have to be a done a little different. With any problem involving fractions or division, look at all the denominators (numbers on bottom) and multiply by the least common denominator. This will get rid of all division or fractions in the problem.

 

 

 

 

Example

 

= 2

 

Since there is one fraction, the LCD will be its denominator, which is 5. Multiply every term on both sides by 5.

(5) = 2(5)

 

The 5’s on the left cancel and 2(5) = 10

 

y + 2 = 10

 

Subtract 2 from both sides.

y = 8

 

Answer!

 

 

 

 

 

 

Example

 

3(a – 2) = -21

 

Get rid of all parentheses by first distributing the 3.

 

3a – 6 = -21

 

Add 6 to both sides.

3a = -15

Divide both sides by 3.

a = -5

 

Answer!

 

 

Here are some further examples of problems using combinations of the previous topics.

 

 

Example

 

= - 15

 

Multiply both sides by –2, this will get the k term by itself as well as get rid of all fractions.

 

(- 2) = - 15(-2)

 

The –2’s on the left side cancel and –15(-2)= 30.

 

3k = 30

 

Divide both sides by 3 to get k by itself.

 

=

The 3’s on the left cancel and = 10.

k = 10

 

Answer!

 

 

 

 

Example

 

3(2x –5) = -27

 

Distribute the 3 to get rid of the parentheses.

6x – 15 = -27

Add 15 to both sides.

6x = -12

 

Divide both sides by 6.

 

x = - 2

 

Answer!

 

 

 

Example

 

= -1

 

Multiply both sides by 3, to get rid of all the parentheses.

 

(3) = -1(3)

 

The 3’s cancel out, and –1(3)= -3.

 

-4x + 5 = -3

 

Now, subtract 5 from both sides.

-4 x = -8

 

Divide both sides by –4.

 

x = 2

 

Answer!

 

The reason all of the previous problems fell under the category of “linear” is because the highest exponent with the variable in each case is 1.