Question

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

$\left\{\begin{array}{l}6x+5y=7\\ 2x+2y=2\end{array}\right.$

Short answer

The solution for the system of equations $\left\{\begin{array}{l}6x+5y=7\\ 2x+2y=2\end{array}\right.$is,$x=2,y=-1$.

Step-by-step solution

Step 1. Given information 

$\left\{\begin{array}{l}6x+5y=7\\ 2x+2y=2\end{array}\right.$

The inverse found in the problem$33$is,$\left[\begin{array}{cc}1& -\frac{5}{2}\\ -1& 3\end{array}\right]$.

Step 2. We know that, X=A-1.B

From the given information, $B=\left[\begin{array}{c}7\\ 2\end{array}\right]$.

Substitute the known values in the formula.

role="math" localid="1647425653982" $$\begin{gathered} \left[\begin{array}{c}x\\ y\end{array}\right]=X \\ X=\left[\begin{array}{cc}1& -\frac{5}{2}\\ -1& 3\end{array}\right]\left[\begin{array}{c}7\\ 2\end{array}\right] \\ =\left[\begin{array}{c}1\left(7\right)-\frac{5}{2}\left(2\right)\\ -1\left(7\right)+3\left(2\right)\end{array}\right] \\ =\left[\begin{array}{c}7-5\\ -7+6\end{array}\right] \\ =\left[\begin{array}{c}2\\ -1\end{array}\right] \end{gathered}$$

So, the solution is, in ordered pairs$\left(2,-1\right)$or$x=2,y=-1$.