Chapter 6.1, Problem 27PPE

To determine

To graph: the system of equations $y-x=5,3y=3x+15$

Explanation of Solution

Given information: The system of equations is $y-x=5,3y=3x+15$

Graph:

First find the points that can be used to plot the graphs of equations $y-x=5,3y=3x+15$

For $y-x=5$ :

This equation can be written as $y-x=5\Rightarrow y=x+5$

  $\begin{array}{l}x=0;y=0+5=5\\ x=1;y=1+5=6\\ x=-1;y=-1+5=4\\ x=-2;y=-2+5=3\end{array}$

Plot the points $(0,5),(1,6),(-1,4),(-2,3)$

For $3y=3x+15$ :

This equation can be written as $3y=3x+15$

At $x=0$ :

  $3y=0+15\Rightarrow 3y=15\Rightarrow y=5$

At $x=1$ :

  $3y=3+15\Rightarrow 3y=18\Rightarrow y=6$

At $x=-1$ :

  $3y=-3+15\Rightarrow 3y=12\Rightarrow y=4$

At $x=-2$ :

  $3y=-6+15\Rightarrow 3y=9\Rightarrow y=3$

Plot the points $(0,5),(1,6),(-1,4),(-2,3)$

  

The system of linear equations has infinite solutions if the graphs of the two lines represented by the equations are coincident.

From the graph drawn above, it can be observed that the graphs of the equations $y-x=5,3y=3x+15$ are coincident.

So,

The system of equations $y-x=5,3y=3x+15$ has infinite solutions.

One of the solution is $\left(-2,3\right)$ as it lies on the graph of both the equations $y-x=5,3y=3x+15$ .