Chapter 6.1, Problem 29PPE

To determine

To graph: the given system of equations.

Explanation of Solution

Given information: Equations are $2y=x-2,3y=\frac{3}{2}x-3$

Graph:

Find the points that can be used to draw the graph of each of the equations.

For $2y=x-2$ :

This equation can be written as $y=\frac{x-2}{2}$

  $\begin{array}{l}x=0;y=\frac{-2}{2}=-1\\ x=2;y=\frac{2-2}{2}=0\\ x=4;y=\frac{4-2}{2}=1\\ x=6;y=\frac{6-2}{2}=2\end{array}$

Plot the points $(0,-1),(2,0),(4,1),(6,2)$

For $3y=\frac{3}{2}x-3$ :

This equation can be written as $y=\frac{x}{2}-1$

  $\begin{array}{l}x=0;y=0-1=-1\\ x=2;y=\frac{2}{2}-1=1-1=0\\ x=4;y=\frac{4}{2}-1=2-1=1\\ x=6;y=\frac{6}{2}-1=3-1=2\end{array}$

Plot the points $(0,-1),(2,0),(4,1),(6,2)$

  

Here, the lines representing equations $2y=x-2,3y=\frac{3}{2}x-3$ are coincident.

So, the system of equations has infinite solutions.

(If the lines representing system of equations are coincident then the system has infinite solutions)

One of the solutions is $(0,-1)$ .