Adapted from Contemporary GED Math by McGraw Hill p 294-301
Solving Algebraic Equations
An equation states that two amounts are equal. The amounts are separated by the equal (=)
sign.
Think of a balance. If you add weight to one side you must add weight to the other to get it to balance.
If you take weight off of one side you must also take weight off of the other side if you want it to
balance. For example, if you have one apple on each side of a balance and you take an apple off of one
side but do nothing to the other side then it no longer balances. Algebra works the same way.
Everything you do to one side of the equation you must also do to the other to keep it balanced.
Let’s look at the equation 1=1. If you add 2 to one side of the equal sign you get 1+2=1. This
statement is not true. If you want the statement to remain true, you must add 2 to both sides of
the equation (1+2=1 +2 or 3=3).
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When you balance equations, it is important to know the inverse (or opposite) operation.
Addition and subtraction are opposites.
Multiplication and division are opposites.
Exponents and roots are opposites.
Adapted from Contemporary GED Math by McGraw Hill p 294-301
Steps to solving a one step equation:
1. The first step to balancing a simple equation is to isolate the variable (get it alone).
For example: x + 23= 43. Here we want to get the x on one side of the equal sign and
all the other numbers on the other. Using a table like this can be helpful.
x + 23 = 43
-23
x
-23
= 20
To get the variable alone you must use the opposite operation, in this
case the opposite of addition is subtraction, so I will subtract 23 from
each side to get only x left on one side of the equal sign and 20 left on the
other. So the answer to this problem is x=20. To check you can substitute 20 for x and
see that 20 + 23=43.
Steps to solving an equation with more than one step
1. Add or subtract
2. Multiply or divide
(NOTE: This is the opposite of PEMDAS)
2x + 6 = 12
-6 -6
2x = 6
2
2
x = 3
To solve this problem, you start by subtracting 6 from each side (remember
subtracting is the opposite of adding, so since the problem says +6, you will instead
subtract 6 from each side. The +6 and the -6 cancel each other out so then you are
left with 2x= 6. The next step is to get the x alone so now divide each side by 2.
2/2=1 so you are left with x on the left and 6/2=3, so you are left with 3 on the right.
Therefore, x=3. Check your answer by substituting the 3 for x in the original
equation.
Combining like terms
Sometimes there is more than of the same variable or more than one number
without a variable. To deal with this you need to combine like terms. This means
you need to get all of the terms that are the same together. For example, if an
equation says x + x +2+3=5, you first need to combine the x + x to make 2x and
Adapted from Contemporary GED Math by McGraw Hill p 294-301
combine 2+3 to get 5. Now the equation looks like this: 2x + 5 = 5. You already
know how to solve this.
Here is a review of the steps needed to solve when variables are separated.
1. Combine the unknowns (if the variables are on different sides of the equal sign
then combine them to the side that has the most variables to avoid dealing with
negative numbers.
2. Use opposite operations to solve the equation
Solving equations with parentheses
1. When an equation contains parentheses, use the Distributive Property to multiply
the variable or number in front of the parentheses by each term inside the
parentheses. For example, 6(3 + x) would become 18 + 6x.
2. Combine like terms.
3. Solve using opposite operations