Lesson 1: Understand Equations and Solutions

Property

An equation is a statement that two expressions are equal. It may involve one or more variables. A value of the variable that makes an equation true is called a solution of the equation, and the process of finding this value is called solving the equation.

Examples

• The statement $x + 5 = 12$ is an equation. The value $x=7$ is a solution because $7 + 5 = 12$ is a true statement.
• To check if $y=3$ is a solution to $8y = 24$, we substitute it in: $8(3) = 24$. This is true, so $y=3$ is a solution.
• Is $z=10$ a solution for $z - 4 = 5$? We check: $10 - 4 = 6$. Since $6$ is not equal to $5$, $z=10$ is not a solution.

Explanation

Think of an equation as a perfectly balanced scale. A solution is the specific value for the variable that keeps the scale level. Finding that value is what we call solving the equation.

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.