Lesson 7: Checking Whether a Value Makes an Equation True

Property

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

To determine whether a number is a solution to an equation.
Step 1. Substitute the number in for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

Examples

• Is $y=4$ a solution to $5y - 3 = 17$? Substitute $y=4$: $5(4) - 3 = 20 - 3 = 17$. Since $17=17$, yes, it is a solution.
• Is $x=3$ a solution to $2x + 8 = x+4$? Substitute $x=3$: $2(3) + 8 = 14$ and $3+4=7$. Since $14 \neq 7$, it is not a solution.
• Is $a = \frac{1}{2}$ a solution to $8a - 1 = 3$? Substitute $a=\frac{1}{2}$: $8(\frac{1}{2}) - 1 = 4 - 1 = 3$. Since $3=3$, yes, it is a solution.

Property

An equation is a statement that two mathematical expressions are equal. This statement can be either true (if the values on both sides of the equal sign are the same) or false (if the values are different). For an equation with a variable, the equation is true only for specific values of that variable, which are called solutions.

Examples

• The equation $5 + 3 = 8$ is a true statement because both sides equal 8.
• The equation $4 \times 3 = 10$ is a false statement because the left side equals 12, and $12 \neq 10$.
• For the equation $x + 5 = 9$, the statement is true only when $x = 4$. If you substitute any other value for $x$, the statement becomes false.

Explanation

Think of an equation as a question: "Are the two sides really equal?" If they are, the equation is true. If they are not, the equation is false. When an equation includes a variable, solving the equation means finding the specific value for the variable that makes the equation a true statement.