Question

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

$\left\{\begin{array}{l}2x+y=0\\ x+y=5\end{array}\right.$

Short answer

The solution for the system of equations $\left\{\begin{array}{l}2x+y=0\\ x+y=5\end{array}\right.$is,$x=-5,y=10$.

Step-by-step solution

Step 1. Given information

$\left\{\begin{array}{l}2x+y=0\\ x+y=5\end{array}\right.$

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, $B=\left[\begin{array}{c}0\\ 5\end{array}\right]$.

Substitute the known values in the formula.

$$\begin{gathered} \left[\begin{array}{c}x\\ y\end{array}\right]=X \\ X=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\left[\begin{array}{c}0\\ 5\end{array}\right] \\ =\left[\begin{array}{c}1\left(0\right)-1\left(5\right)\\ -1\left(0\right)+2\left(5\right)\end{array}\right] \\ =\left[\begin{array}{c}0-5\\ 0+10\end{array}\right] \\ =\left[\begin{array}{c}-5\\ 10\end{array}\right] \end{gathered}$$

So, the solution is, in ordered pairs$\left(-5,10\right)$or$x=-5,y=10$.