Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}3x-4y=4\\ \frac{1}{2}x-3y=-\frac{1}{2}\end{array}\right. \\ x=2, y=\frac{1}{2};\left(2,\frac{1}{2}\right) \end{gathered}$$

Step-by-Step Solution

Answer

The given values are the solutions because both equations are true for the given values.

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}3x-4y& =& 4\\ \frac{1}{2}x-3y& =& -\frac{1}{2}\end{array}$

The given solution is $x=2, y=\frac{1}{2}$ or $\left(2,\frac{1}{2}\right)$.

Step 2. Check the solution.

If both equations are true for the given values, then the values represent the solution.

Substitute $x=2, y=\frac{1}{2}$ in the first equation.

$\begin{array}{rcl}3\left(2\right)-4\left(\frac{1}{2}\right)& =& 4\\ 6-2& =& 4\\ 4& =& 4\end{array}$

This statement is true.

Substitute $x=2, y=\frac{1}{2}$ in the second equation.

$\begin{array}{rcl}\frac{1}{2}\left(2\right)-3\left(\frac{1}{2}\right)& =& -\frac{1}{2}\\ 1-\left(\frac{3}{2}\right)& =& -\frac{1}{2}\\ -\frac{1}{2}& =& -\frac{1}{2}\end{array}$

This statement is true.

Step 3. Conclusion.

The given values are the solutions because both equations are true for the given values.