Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

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 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.

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

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