Chapter 6.1, Problem 30PPE

To determine

To graph: the system of equations $3x-y=2,4y=-x+5$

Explanation of Solution

Given information: The system of equations is $3x-y=2,4y=-x+5$

Graph:

First find the points that can be used to plot the graphs of equations $3x-y=2,4y=-x+5$

For $3x-y=2$ :

This equation can be written as $3x-y=2\Rightarrow y=3x-2$

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

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

For $4y=-x+5$ :

This equation can be written as $4y=-x+5\Rightarrow y=\frac{-x+5}{4}$

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

Plot the points $\left(0,\frac{5}{4}\right),\left(1,1\right),\left(2,\frac{3}{4}\right),\left(-3,2\right)$

  

The system of linear equations has unique solution if the graphs of the two lines represented by the equations are intersecting.

From the graph drawn above, it can be observed that the graphs of the equations $3x-y=2,4y=-x+5$ intersect at point $\left(1,1\right)$ .

So,

The system of equations $3x-y=2,4y=-x+5$ has solution as $\left(1,1\right)$ .