Is a Number a Solution of an Equation?
When some people hear the word algebra, they think of solving equations. The applications of solving equations are limitless and extend to all careers and fields. In this section, we will begin solving equations. We will start by solving basic equations, and then as we proceed through the course we will build up our skills to cover many different forms of equations.
Determining 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.
\(\begin{array}{}\hfill x+2=7\hfill \\ \hfill 5+2\stackrel{?}{=}7\hfill \\ \\ \hfill \phantom{\rule{3em}{0ex}}7=7✓\hfill \end{array}\)
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 \(\text{(=)}\) or not-equal sign \(\text{(≠).}\)
Determining whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- 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
\(\text{Determine whether}\phantom{\rule{0.2em}{0ex}}x=5\phantom{\rule{0.2em}{0ex}}\text{is a solution of}\phantom{\rule{0.2em}{0ex}}6x-17=16.\)
Solution
| Multiply. | |
| Subtract. |
So \(x=5\) is not a solution to the equation \(6x-17=16.\)
Example
\(\text{Determine whether}\phantom{\rule{0.2em}{0ex}}y=2\phantom{\rule{0.2em}{0ex}}\text{is a solution of}\phantom{\rule{0.2em}{0ex}}6y-4=5y-2.\)
Solution
Here, the variable appears on both sides of the equation. We must substitute \(2\) for each \(y.\)
| Multiply. | |
| Subtract. |
Since \(y=2\) results in a true equation, we know that \(2\) is a solution to the equation \(6y-4=5y-2.\)
This lesson is part of:
The Language of Algebra