## Determining Whether a Number is a Solution to an Equation

### Learning Outcomes

• Determine whether a whole number is a solution to an equation
• Determine whether an integer is a solution to an equation

## Determine Whether a Number is a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!

### Solution of an Equation

A solution to an equation is a value of a variable that makes a true statement when substituted into the equation.
The process of finding the solution to an equation is called solving the equation.

To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of $x+2=7?$ If you said $5$, you’re right! We say $5$ is a solution to the equation $x+2=7$ because when we substitute $5$ for $x$ the resulting statement is true.

$\begin{array}{}\\ \hfill x+2=7\hfill \\ \hfill 5+2\stackrel{?}{=}7\hfill \\ \\ \hfill 7=7\quad\checkmark \hfill \end{array}$

Since $5+2=7$ is a true statement, we know that $5$ is indeed a solution to the equation.

The symbol $\stackrel{?}{=}$ asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign $=$ or not-equal sign $\not=$.

### Determine whether a number is a solution to an equation.

1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
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.

### example

Determine whether $x=5$ is a solution of $6x - 17=16$.

Solution

 $6x-17=16$ Substitute $\color{red}{5}$ for x. $6\cdot\color{red}{5}-17=16$ Multiply. $30-17=16$ Subtract. $13\not=16$

So $x=5$ is not a solution to the equation $6x - 17=16$.

### example

Determine whether $y=2$ is a solution of $6y - 4=5y - 2$.

## Determine Whether an Integer is a Solution of an Equation

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

### example

Determine whether each of the following is a solution of $2x - 5=-13\text{:}$

1. $x=4$
2. $x=-4$
3. $x=-9$

Solution

 1. Substitute $4$ for x in the equation to determine if it is true. $2x-5=-13$ Substitute $\color{red}{4}$ for x. $2(\color{red}{4})-5=-13$ Multiply. $8-5=-13$ Subtract. $3\not=-13$

Since $x=4$ does not result in a true equation, $4$ is not a solution to the equation.

 2. Substitute $−4$ for x in the equation to determine if it is true. $2x-5=-13$ Substitute $\color{red}{-4}$ for x. $2(\color{red}{-4})-5=-13$ Multiply. $-8-5=-13$ Subtract. $-13=-13\quad\checkmark$

Since $x=-4$ results in a true equation, $-4$ is a solution to the equation.

 3. Substitute $−9$ for x in the equation to determine if it is true. $2x-5=-13$ Substitute $−9$ for x. $2(\color{red}{-9})-5=-13$ Multiply. $-18-5=-13$ Subtract. $-23\not=-13$

Since $x=-9$ does not result in a true equation, $-9$ is not a solution to the equation.

### try it

In the following video we show more examples of how to verify whether an integer is a solution to a linear equation.