# How Do You Write Out and Solve an Equation from an Application?

Linear models have inputs and outputs. In applications, the input or output may be specified and you must solve for the other variable. To make sure you put in the value for the correct variable, pay careful attention to what each variable represents.

Once the resulting equation is solved, you can check it with the Method of Intersection. In this method, each side of the equation is graphed to find the point of intersection. The horizontal value of the point of intersection should match the algebraic solution if everything has been done correctly.

Example  An antique is originally purchased for \$20,000. The value V after t months is $\displaystyle V=1500t+20,000$ dollars. After how many months will the value be \$36,000?

Solution To find when the value will be \$36,000, set V equal to 36,000 and solve for t. Verify the algebraic solution by graphing each side of the resulting equation. Since the linear model starts at 20,000 and increases to 36,000, an appropriate window is [0,12] by [0, 40000]. Other windows like [0,20] by [10000, 40000] would also work well.