Solving Equations
Solving Equations with Addition and Subtraction
You can add or subtract the same number on each side of an
equation and the result is an equivalent equation
. Equivalent equations have the same solution.
| Addition Property of Equality |
For any numbers a, b, and c, if a = b,
then a + c = b + c. |
| Subtraction Property of Equality |
For any numbers a, b, and c, if a = b,
then a - c = b - c. |
| Solving Equations |
To solve an equation
means to get the variable (with a coefficient of 1) by
itself on one side of the equation. You can do this by
undoing what has been done to the variable, using the
properties of equality. |
Example
Solve  .
Solution
The number  has been subtracted from x. The opposite
of subtracting is adding .
Add to each side of the equation. 
is an equivalent equation. Simplify to obtain x = 1.
Check: Is  ? Yes. The solution is 1.
Solving Equations with Multiplication and Division
You can solve a multiplication or division equation by using
the multiplication and division properties of equality.
| Multiplication Property of
Equality |
For any numbers a, b, and c, if a = b,
then ac = bc. |
| Division Property of Equality |
For any numbers a, b, and c, with c  0, if a = b, then  . |
Example
Solve 
Solution
Rewrite the mixed numbers as improper fractions.
 Multiply each side by  ,
the reciprocal of the number that is multiplied by x.
Solving Multi-Step Equations
| Solving Multi-Step Equations |
- Work backward to isolate the variable and solve
the equation.
- Use subtraction to undo addition, and use
addition to undo subtraction.
- Use multiplication to undo division, and use
division to undo multiplication.
|
Consecutive integers are integers in counting
order, such as -3, -2, and -1.
Example
Solve  .
Solution
Multiply each side by 4 to eliminate the fraction.
Next, undo the subtraction by adding 3 to each side.
2x - 3 + 3 = 36 + 3
2x = 39
Last, undo the multiplication by dividing each side by 2.
|
