Chapter 6.1, Problem 23PPE

To determine

To graph: the system of equations $y=2x-1,3y=6x-5$

Explanation of Solution

Given information: The system of equations is $y=2x-1,3y=6x-5$

Graph:

First find the points that can be used to plot the graphs of equations $y=2x-1,3y=6x-5$

For $y=2x-1$ :

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

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

For $3y=6x-5$ :

This equation can be written as $y=\frac{6x-5}{3}\Rightarrow y=2x-\frac{5}{3}$

  $\begin{array}{l}x=0;y=0-\frac{5}{3}=-\frac{5}{3}\\ x=1;y=2-\frac{5}{3}=\frac{1}{3}\\ x=2;y=4-\frac{5}{3}=\frac{7}{3}\\ x=3;y=6-\frac{5}{3}=\frac{13}{3}\end{array}$

Plot the points $\left(0,\frac{-5}{3}\right),\left(1,\frac{1}{3}\right),\left(2,\frac{7}{3}\right),\left(3,\frac{13}{3}\right)$

  

The system of linear equations has no solution if the graphs of the two lines represented by the equations are parallel.

From the graph drawn above, it can be observed that the graphs of the equations $y=2x-1,3y=6x-5$ are parallel.

So,

The system of equations $y=2x-1,3y=6x-5$ has no solution.