Chapter 6.1, Problem 32PPE

To determine

To find: student’s error in concluding that the given system of equations has no solution.

Explanation of Solution

Given information: Graph for the system of equations $y=-x+3,y=-2x-1$ is given

  

In the graph above, student does not extent the lines representing each of the equations further. So, it seems that the lines are parallel.

And we know that if the lines representing each of the equations are parallel then the system of equations has no solution.

First draw the graph again.

Find points that can be used to plot the graph:

For $y=-x+3$ :

  $\begin{array}{l}x=0;y=0+3=3\\ x=1;y=-1+3=2\\ x=2;y=-2+3=1\\ x=3;y=-3+3=0\\ x=-4;y=4+3=7\end{array}$

Plot the points $(0,3),(1,2),(2,1),(3,0),(-4,7)$

For $y=-2x-1$ :

  $\begin{array}{l}x=0;y=0-1=-1\\ x=1;y=-2-1=-3\\ x=-2;y=4-1=3\\ x=-1;y=2-1=1\\ x=-4;y=8-1=7\end{array}$

Plot the points $(0,-1),(1,-3),(-2,3),(-1,1),(-4,7)$

  

In the graph it can be observed that the lines are intersecting at point $(-4,7)$

We know that if two lines are intersecting then the system of equations has unique solution. In this case, the solution is $(-4,7)$ .