Q. 45 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

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 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.

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

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