Chapter 5.1, Problem 15E

To determine

To solve: the system of linear equation.

Answer to Problem 15E

All of the equalities are true; the ordered pair $\left(-9,-1\right)$ is the solution of the system.

Explanation of Solution

Given:

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

Calculation:

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

Since both expressions $\frac{1}{3}x+2$ and $\frac{2}{3}x+5$ are equal to y

Set them equal to each other forming an equation in x

  $\frac{1}{3}x+2=\frac{2}{3}x+5$

Multiply both sides of equations by 3,

  $x+6=2x+15$

Move variable 2x to the left side and change its sign

  $\begin{array}{l}x-2x+6=15\\ x-2x=15-6\end{array}$

Collect like terms

  $-x=9$

Divide both sides by -1

  $x=-9$

Substitute the value of x into the equation

  $\begin{array}{l}y=\frac{1}{3}x+2\\ y=\frac{1}{3}\times \left(-9\right)+2\end{array}$

Solve the equation for y

  $y=-1$

The possible solution of the system is the ordered pair $\left(x,y\right)=\left(-9,-1\right)$

Check if the given ordered pair is the solution of the system of equations

  $\begin{array}{l}-1=\frac{1}{3}\times \left(-9\right)+2\\ -1=\frac{2}{3}\times \left(-9\right)+5\end{array}$

Simplify the equalities

  $\begin{array}{l}-1=-1\\ -1=-1\end{array}$

Since all of the equalities are true, the ordered pair $\left(-9,-1\right)$ is the solution of the system.

  

Conclusion:

Therefore, all of the equalities are true; the ordered pair $\left(-9,-1\right)$ is the solution of the system.