Chapter 8.8, Problem 24E

To determine

To find:

The coordinates of the triangle’s vertices.

Answer to Problem 24E

The coordinates of the triangle’s vertices are $(3,2)$ , $(-3,6)$ and $(0,-4)$ .

Explanation of Solution

Given:

The graphs of the three equations below form a triangle.

  $\begin{array}{l}2x-y=\mathrm{4...}(1)\\ 2x+3y=\mathrm{12...}(2)\\ 10x+3y=-\mathrm{12...}(3)\end{array}$

Calculation:

First of all, we will solve equations (1) and (2) using substitution method.

  $\begin{array}{l}2x-y+y=4+y\\ 2x=4+y\end{array}$

Upon substituting this value in equation (2), we will get:

  $\begin{array}{l}2x+3y=12\\ 4+y+3y=12\left(\text{Substituting }2x=4+y\right)\\ 4y+4=12\left(\text{Combining like terms}\right)\\ 4y=8\left(\text{Subtracting 4 from both sides}\right)\\ y=2\left(\text{Dividing both sides by 4}\right)\end{array}$

Now, we will substitute $y=2$ in equation (1) as:

  $\begin{array}{l}2x-y=4\\ 2x-2=4\left(\text{Substituting }y=2\right)\\ 2x-2+2=4+2\left(\text{Adding 2 on both sides}\right)\\ 2x=6\\ x=3\left(\text{Dividing both sides by 2}\right)\end{array}$

Therefore, one vertex of triangle will be $(3,2)$ .

Now, we will solve equations (2) and (3) using substitution method.

  $\begin{array}{l}2x+3y=12\\ 2x-2x+3y=12-2x\\ 3y=12-2x\end{array}$

Upon substituting this value in equation (3), we will get:

  $\begin{array}{l}10x+3y=-12\\ 10x+12-2x=-12\left(\text{Substituting }3y=12-2x\right)\\ 8x+12=-12\left(\text{Combining like terms}\right)\\ 8x+12-12=-12-12\left(\text{Subtracting 12 from both sides}\right)\\ 8x=-24\\ x=-3\left(\text{Dividing both sides by 8}\right)\end{array}$

Now, we will substitute $x=-3$ in equation (2) as:

  $\begin{array}{l}2x+3y=12\\ 2(-3)+3y=12\left(\text{Substituting }x=-3\right)\\ -6+3y=12\\ -6+6+3y=12+6\left(\text{Adding 6 on both sides}\right)\\ 3y=18\\ y=6\left(\text{Dividing both sides by 3}\right)\end{array}$

Therefore, second vertex of triangle will be $(-3,6)$ .

Let us solve equations (1) and (3) using substitution method.

  $\begin{array}{l}2x-y+y=4+y\\ 2x=4+y\\ y=2x-4\end{array}$

Upon substituting this value in equation (3), we will get:

  $\begin{array}{l}10x+3y=-12\\ 10x+3(2x-4)=-12\left(\text{Substituting }y=2x-4\right)\\ 10x+6x-12=-12\\ 16x-12=-12\left(\text{Combining like terms}\right)\\ 16x-12+12=-12+12\left(\text{Adding 12 on both sides}\right)\\ 16x=0\\ x=0\left(\text{Dividing both sides by 16}\right)\end{array}$

Now, we will substitute $x=0$ in equation (1) as:

  $\begin{array}{l}2x-y=4\\ 2(0)-y=4\\ 0-y=4\\ -y=4\\ y=-4\end{array}$

Therefore, second vertex of triangle will be $(0,-4)$ .