Chapter 3, Problem 11SGR

To determine

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

  $\begin{array}{c}3x+4y=8\\ x-3y=-6\end{array}$

Answer to Problem 11SGR

The solution set for the system of equations is $\left(0,2\right)$ .

Explanation of Solution

Given information:

The system of equations

  $\begin{array}{c}3x+4y=8\\ x-3y=-6\end{array}$

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

  $\begin{array}{c}3x+4y=8\\ x-3y=-6\end{array}$

Take the first equation $3x+4y=8$ and write it in slope intercept form.

Subtract both sides by $3x$ ,

  $4y=8-3x$

Divide both sides by 4,

  $\begin{array}{c}y=\frac{8-3x}{4}\\ y=\frac{8}{4}-\frac{3x}{4}\\ y=-\frac{3}{4}x+2\end{array}$

Take the second equation $x-3y=-6$ and write it in slope intercept form.

Subtract both sides by $x$ ,

  $\begin{array}{c}-3y=-6-x\\ 3y=6+x\end{array}$

Divide both sides by 3,

  $\begin{array}{c}-3y=-6-x\\ y=\frac{6+x}{3}\\ y=\frac{6}{3}+\frac{x}{3}\\ y=\frac{1}{3}x+2\end{array}$

Plot the equations $y=-\frac{3}{4}x+2$ and $y=\frac{1}{3}x+2$ on the coordinate plane.

  

Red line depicts the equation $y=-\frac{3}{4}x+2$ and blue line depicts the equation $y=\frac{1}{3}x+2$ .

The graph of the equations intersect at $\left(0,2\right)$ .

Therefore, the solution set of the system of equations is $\left(0,2\right)$ .

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

For the equation $3x+4y=8$ , substitute x as 0 and y as 2.

  $\begin{array}{c}3\left(0\right)+4\left(2\right)=8\\ 8=8\end{array}$

For the equation $x-3y=-6$ , substitute x as 0 and y as 2.

  $\begin{array}{c}0-3\left(2\right)=-6\\ -6=-6\end{array}$

Both the equations satisfy the point.

Thus, the solution set for the system of equations is $\left(0,2\right)$ .