Chapter 6.2, Problem 8P

Solve each system by substitution. Check your solution.

$6x-4y=54$

$-9x+2y=-69$

To determine

To solve: The system by substitution and check the solution.

  $\begin{array}{l}6x-4y=54\\ -9x+2y=-69\end{array}$

Answer to Problem 8P

Therefore the solution is $\left(x,y\right)=(7,-3)$

Explanation of Solution

Given information: Solve these two system by substitution

  $\begin{array}{l}6x-4y=54\\ -9x+2y=-69\end{array}$

Calculation: In order to solve these above equation, we solve one of the equations for one of the Variables, and plug this into other equation.

  $\begin{array}{l}6x-4y=54\\ -9x+2y=-69\end{array}$

First we solve equation $6x-4y=54$ as follows:

  $\begin{array}{l}6x-4y=54\\ \Rightarrow 6x=54+4y\\ \Rightarrow x=\frac{54}{6}+\frac{4}{6}y\\ \Rightarrow x=9+\frac{2}{3}y\end{array}$

Now we substitute $x=9+\frac{2}{3}y$ into $-9x+2y=-69$ , we get,

  $\begin{array}{l}-9x+2y=-69\\ \Rightarrow -9\left(9+\frac{2}{3}y\right)+2y=-69\\ \Rightarrow -81-6y+2y=-69\\ \Rightarrow -4y=12\\ \Rightarrow y=-3\end{array}$

By plugging y =-3, we get

  $\begin{array}{l}x=9+\frac{2}{3}y\\ \Rightarrow x=9+\frac{2}{3}(-3)\\ \Rightarrow x=9-2\\ \Rightarrow x=7\end{array}$

Therefore the solution is

  $\left(x,y\right)=(7,-3)$

Verification:

In order to check it, we replace $\left(x,y\right)=(7,-3)$ in any one system, we get

  $\begin{array}{l}6x-4y=54\\ \Rightarrow 6\times 7-4(-3)=54\\ \Rightarrow 42+12=54\\ \Rightarrow 54=54\end{array}$