Chapter 3, Problem 13SGR

To determine

To calculate: The solution set for the system of equations with the help of a graph.

  $\begin{array}{c}y-3x=13\\ y=\frac{1}{3}x+5\end{array}$

Answer to Problem 13SGR

The solution set for the system of equations is $\left(-3,4\right)$ .

Explanation of Solution

Given information:

The system of equations

  $\begin{array}{c}y-3x=13\\ y=\frac{1}{3}x+5\end{array}$

Formula used:

The slope intercept of a line is $y=mx+c$ where m is the slope and c is the intercept of y -axis.

Calculation:

Consider the system of equations

  $\begin{array}{c}y-3x=13\\ y=\frac{1}{3}x+5\end{array}$

Take the first equation $y-3x=13$ and write it in slope intercept form.

Add both sides $3x$ ,

  $\begin{array}{c}y=13+3x\\ y=3x+13\end{array}$

Take the second equation $y=\frac{1}{3}x+5$ it is already in slope intercept form.

Plot the equations $y=3x+13$ and $y=\frac{1}{3}x+5$ on the coordinate plane.

  

Red line depicts the equation $y=3x+13$ and blue line depicts the equation $y=\frac{1}{3}x+5$ .

The graph of the equations intersect at $\left(-3,4\right)$ .

Therefore, the solution set of the system of equations is $\left(-3,4\right)$ .

To verify substitute the point $\left(0,2\right)$ in the equations and check.

For the equation $y-3x=13$ , substitute x as $-3$ and y as 4.

  $\begin{array}{c}4-3\left(-3\right)=13\\ 4+9=13\\ 13=13\end{array}$

For the equation $y=\frac{1}{3}x+5$ , substitute x as $-3$ and y as 4.

  $\begin{array}{c}4=\frac{1}{3}\left(-3\right)+5\\ 4=-1+5\\ 4=4\end{array}$

Both the equations satisfy the point.

Thus, the solution set for the system of equations is $\left(-3,4\right)$ .