Question

In Problems $41-60$, use the inverses found in Problems 31–40 to solve each system of equations.

$\left\{\begin{array}{l}2x+y=8\\ x+y=5\end{array}\right.$

Short answer

The solution of the system of equations $\left\{\begin{array}{l}2x+y=8\\ x+y=5\end{array}\right.$is,$\left(3,2\right)$.

Step-by-step solution

Step 1. Given information

$\left\{\begin{array}{l}2x+y=8\\ x+y=5\end{array}\right.$

The inverse found in problem$31$is,$\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$

Step 2. We know that, X=A-1.B.

From the given information, $B=\left[\begin{array}{c}8\\ 5\end{array}\right]$

Substitute the known values in the formula.

role="math" localid="1647416830279" $$\begin{gathered} X=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\left[\begin{array}{c}8\\ 5\end{array}\right] \\ =\left[\begin{array}{c}\left(1\times 8\right)-1\left(5\right)\\ \left(-1\times 8\right)+10\end{array}\right] \\ =\left[\begin{array}{c}8-5\\ -8+10\end{array}\right] \\ =\left[\begin{array}{c}3\\ 2\end{array}\right] \end{gathered}$$

So, the obtained solution is,$x=3,y=2$or using ordered pair$\left(3,2\right)$.