When businesspeople use algebra to make important decisions such as how much to charge for a product, they start by setting up an equation—a statement that say two expressions are equal. Then they solve for the unknown quantity.
In other words, they find a solution--a value that they can put in place of a variable (such as x) that makes the equation true. Any time you need to find the solution to an equation, one or more of the variables in the equation will be designated as an unknown.
When you solve an equation, your first step is always to isolate the variable, which means you have to get the variable by itself on one side of the equation. And to isolate a variable, you must “undo” what is being done to it.
Inverse Operations
To undo what is being done to a variable, you use inverse operations—mathematical operations that undo each other. For example, addition undoes subtraction, and multiplication undoes division. Likewise, subtraction undoes addition, and division undoes multiplication, and vice versa.
Here is an important rule to remember: You can perform the same inverse operation on each side of an equation without changing the equality.
Properties of Equalities
The properties that deal with inverse operations are called the properties of equality. Because properties of equalities hold true for all equations, they allow you to balance, manipulate, and solve equations.
Let's Break it Down
Study the explanations on the tabs below to review the four properties of equality.
The addition property of equality says that if you add the same number to both sides of an equation, the two sides remain equal. In other words, the equality holds true.
\(\mathsf{ \begin{gather} \text{If } a - b = c \\ \text{then } a - b + b = c + b \\ \text{or } a = c + b \end{gather} }\)
For example, suppose you have the equation \(\mathsf{ 5 – 2 = 3 }\). The addition property of equality states that if you add 2 to both sides of the equation, the expressions will still be equal.
\(\mathsf{ \begin{gather} 5 – 2 = 3 \\ 5 – 2 + 2 = 3 + 2 \\ 5 = 5 \end{gather} }\)
Notice how the addition undoes the subtraction.
Like the addition property of equality, the subtraction property of equality states that if you subtract the same number from both sides of an equation, the equality holds true.
\(\mathsf{ \begin{gather} \text{If } a + b = c \\ \text{then } a + b - b = c - b \\ \text{or } a = c - b \end{gather} }\)
For example, if you have the equation \(\mathsf{ 3 + 2 = 5 }\), then the subtraction property of equality states that if you subtract 2 from both sides of the equation, the expressions will still be equal.
\(\mathsf{ \begin{gather} 3 + 2 = 5 \\ 3 + 2 – 2 = 5 – 2 \\ 3 = 3 \end{gather} }\)
Notice, the subtraction undoes the addition.
If you multiply each side of an equation with the same nonzero number, the expressions are still equal to one another.
\(\mathsf{ \begin{gather} \text{If } \frac{a}{b} = c \\ \text{then } \frac{a}{b} \cdot b = c \cdot b \\ \text{or } a = cb \end{gather} }\)
For example, if you have the equation \(\mathsf{ \frac{15}{3} = 5 }\), then the multiplication property of equality states that if you multiply both sides of the equation by 3, the expressions will still be equal.
\(\mathsf{ \begin{gather} \frac{15}{3} = 5 \\ \frac{15}{3} \cdot 3 = 5 \cdot 3 \\ 15 = 15 \end{gather} }\)
Notice how the multiplication undoes the division.
Finally, if you divide both sides of an equation by the same nonzero number, the expressions will still be equal to one another.
\(\mathsf{ \begin{gather} \text{If } a \cdot b \\ \text{then } \frac{a \cdot b}{b} = \frac{c}{b} \\ \text{or } a = \frac{c}{b} \end{gather} }\)
For example, if you have the equation \(\mathsf{ 3 \cdot 5 = 15 }\), then the division property of equality states that if you divide both sides of the equation by 5, the expressions are still equal.
\(\mathsf{ \begin{gather} 3 \cdot 5 = 15 \\ \frac{3 \cdot 5}{5} = \frac{15}{5} \\ 3 = 3 \end{gather} }\)
Notice the division undoes the multiplication.
Think you got it?
Each of the equations in the activity below can be solved in one step, using one property of equality. Match the equation on the left with the property of equality that could be used to isolate the variable (to undo what’s being done to the variable).
