Q. 42 - Precalculus Enhanced with Graphing Utilities

Question

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

$\left\{\begin{array}{l}3x-y=8\\ -2x+y=4\end{array}\right.$

Step-by-Step Solution

Answer

The solution for the system of equations $\left\{\begin{array}{l}3x-y=8\\ -2x+y=4\end{array}\right.$is, $x=12,y=28$.

Step 1 . Given information

$\left\{\begin{array}{l}3x-y=8\\ -2x+y=4\end{array}\right.$

The inverse found in problem $32$ is, $\left[\begin{array}{cc}1& 1\\ 2& 3\end{array}\right]$.

Step 2 . We know that, X = A - 1 . B

From the given information, $B=\left[\begin{array}{c}8\\ 4\end{array}\right]$.

Substitute the known values in the formula.

$$\begin{gathered} \left[\begin{array}{c}x\\ y\end{array}\right]=X \\ X=\left[\begin{array}{cc}1& 1\\ 2& 3\end{array}\right]\left[\begin{array}{c}8\\ 4\end{array}\right] \\ =\left[\begin{array}{c}\left(1\times 8\right)+1\left(4\right)\\ \left(2\times 8\right)+3\left(4\right)\end{array}\right] \\ =\left[\begin{array}{c}8+4\\ 16+12\end{array}\right] \\ =\left[\begin{array}{c}12\\ 28\end{array}\right] \end{gathered}$$

So, the solution is, in ordered pairs $\left(12,28\right)$ or $x=12,y=28$.