Q66. - Algebra 2

Question

Solve each system of equations by using inverse matrices.

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

Step-by-Step Solution

Answer

The solution to the given system of equations is $(1,2)$ .

Step 1. Given Information.

Given to solve the below system of equations by using inverse matrices

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

Step 2. Explanation .

The matrix equation for the system of equations is:$\left[\begin{array}{cc}2& 3\\ 1& -2\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ -3\end{array}\right]$

Where$A=\left[\begin{array}{cc}2& 3\\ 1& -2\end{array}\right]$  , $X=\left[\begin{array}{c}x\\ y\end{array}\right]$  and  $B=\left[\begin{array}{c}8\\ -3\end{array}\right]$

The inverse of coefficient matrix A is given by:$\begin{array}{l}A=\left[\begin{array}{cc}2& 3\\ 1& -2\end{array}\right]\\ A^{-1}=\frac{1}{\left(2\right)\cdot (-2)-\left(3\right)\cdot \left(1\right)}\left[\begin{array}{cc}-2& -3\\ -1& 2\end{array}\right]\\ A^{-1}=\frac{1}{-4-3}\left[\begin{array}{cc}-2& -3\\ -1& 2\end{array}\right]\\ A^{-1}=\frac{1}{-7}\left[\begin{array}{cc}-2& -3\\ -1& 2\end{array}\right]\end{array}$

Multiplying each side of the matrix equation by the inverse matrix:

$\begin{array}{l}\frac{1}{-7}\left[\begin{array}{cc}-2& -3\\ -1& 2\end{array}\right]\cdot \left[\begin{array}{cc}2& 3\\ 1& -2\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{-7}\left[\begin{array}{cc}-2& -3\\ -1& 2\end{array}\right]\cdot \left[\begin{array}{c}8\\ -3\end{array}\right]\\ \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{-7}\left[\begin{array}{c}(-2)\left(8\right)+(-3)(-3)\\ (-1)\left(8\right)+\left(2\right)(-3)\end{array}\right]\\ \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{-7}\left[\begin{array}{c}-16+9\\ -8+-6\end{array}\right]\\ \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{-7}\left[\begin{array}{c}-7\\ -14\end{array}\right]\\ \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]\end{array}$

Step 3. Conclusion .

Hence, the solution to the given system of equations is $(1,2)$ .