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.
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. 
has been subtracted from x. The opposite of subtracting 
is an equivalent equation. Simplify to obtain x = 1.
? Yes. The solution is 1. 
0, if a = b, then 
.
Multiply each side by 
, the reciprocal
of the number that is multiplied by x. 
. 
