Chapter 6.2, Problem 7P

Solve each system by substitution. Check your solution.

$-3x-2y=5.5$

$x+3y=7.5$

To determine

To solve: The system by substitution and check solution.

  $\begin{array}{l}-3x-2y=5.5\\ x+3y=7.5\end{array}$

Answer to Problem 7P

Therefore the solution is

  $\left(x,y\right)=(-4.5,4)$

Explanation of Solution

Given information:

  $\begin{array}{l}-3x-2y=5.5\\ x+3y=7.5\end{array}$

Calculation:

In order to solve two system by substitution, we solve one of the equations for one of the

Variables, and plug this into other equation.

  $\begin{array}{l}-3x-2y=5.5\\ x+3y=7.5\end{array}$

First we solve equation $-3x-2y=5.5$ as follows:

  $\begin{array}{l}-3x-2y=5.5\\ \Rightarrow -3x=5.5+2y\\ \Rightarrow x=-\frac{5.5}{3}-\frac{2}{3}y\end{array}$

Now we substitute $x=-\frac{5.5}{3}-\frac{2}{3}y$ into $x+3y=7.5$ , we get,

  $\begin{array}{l}x+3y=7.5\\ \Rightarrow \left(-\frac{5.5}{3}-\frac{2}{3}y\right)+3y=7.5\\ \Rightarrow -\frac{2}{3}y+3y=7.5+\frac{5.5}{3}\\ \Rightarrow y\left(\frac{-2+9}{3}\right)=\frac{22.5+5.5}{3}\\ \Rightarrow y\frac{7}{3}=\frac{28}{3}\\ \Rightarrow y=4\end{array}$

By plugging $y=4$ in equation $x=-\frac{5.5}{3}-\frac{2}{3}y$ we get

  $\begin{array}{l}x=-\frac{5.5}{3}-\frac{2}{3}y\\ \Rightarrow x=-\frac{5.5}{3}-\frac{2}{3}\times 4\\ \Rightarrow x=\frac{-5.5-8}{3}\\ \Rightarrow x=-4.5\end{array}$

Therefore the solution is

  $\left(x,y\right)=(-4.5,4)$

Verification:

In order to check it, we replace $\left(x,y\right)=(-4.5,4)$ in any one system, we get

  $\begin{array}{l}x+3y=7.5\\ \Rightarrow -4.5+3\times 4=7.5\\ \Rightarrow 7.5=7.5\end{array}$