Chapter 8.8, Problem 20E

To determine

To solve:

The given linear system of equations by graphing.

Answer to Problem 20E

There will be infinitely many solutions for given system of equations.

Explanation of Solution

Given:

A system of equations:

  $\begin{array}{l}3x+2y=8\\ 4y=16-6x\end{array}$

Calculation:

First of all, we will convert our given equations in slope-intercept form as:

  $\begin{array}{l}3x+2y=8\left(\text{Given equation}\right)\\ 3x-3x+2y=8-3x\left(\text{Subtracting }3x\text{ from both sides}\right)\\ 2y=-3x+8\\ \frac{2y}{2}=\frac{-3x}{2}+\frac{8}{2}\left(\text{Dividing both sides by 2}\right)\\ y=-\frac{3}{2}x+4\end{array}$

  $\begin{array}{l}4y=16-6x\left(\text{Given equation}\right)\\ 4y=-6x+16\\ \frac{4y}{4}=-\frac{6x}{4}+\frac{16}{4}\left(\text{Dividing both sides by 4}\right)\\ y=-\frac{3}{2}x+4\end{array}$

We can see that both equations represent same line.

Upon graphing both equations, we will get our required graph as shown below:

  

Upon looking at our graph, we can see that both lines intersect in infinitely many places.Therefore, there will be infinitely many solutions for given system of equations.