Chapter SH, Problem 3.24EP

To determine

To solve: the solution of the given system of equation by using matrix.

Answer to Problem 3.24EP

  $x=-3,y=-2$ .

Explanation of Solution

Given:

The given system of equation is $\begin{array}{l}5x-3y=-9\\ -4x-6y=24\end{array}$ .

Concept used:

By using matrix, the formula of system of linear equation is given below:

  $\begin{array}{l}ifA=\left[\begin{array}{l}ab\\ cd\end{array}\right]\\ then{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{l}d-b\\ -ca\end{array}\right]\end{array}$

Calculation:

The givensystem of equation is $\begin{array}{l}5x-3y=-9\\ -4x-6y=24\end{array}$ .

Therefore, the matrix form of equation is $\left[\begin{array}{l}5-3\\ -4-6\end{array}\right]$

Now, to find the inverse of A so that to find the solution of the system of equation:

  $\begin{array}{l}{A}^{-1}=\frac{1}{5\left(-6\right)-\left(-3\right)\left(-4\right)}\left[\begin{array}{l}-63\\ 45\end{array}\right]\\ =\frac{1}{-30-12}\left[\begin{array}{l}-63\\ 45\end{array}\right]=\frac{1}{-42}\left[\begin{array}{l}-63\\ 45\end{array}\right]\end{array}$

Adjoint of the matrix is given by $X=A^{-1}B$ where $X=\left[\begin{array}{l}x\\ y\end{array}\right]$

  $\begin{array}{l}X=A^{-1}B\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{-42}\left[\begin{array}{l}-63\\ 45\end{array}\right]\left[\begin{array}{l}-9\\ 24\end{array}\right]\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{42}\left[\begin{array}{l}6-3\\ -4-5\end{array}\right]\left[\begin{array}{l}-9\\ 24\end{array}\right]\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{42}\left[\begin{array}{l}6\times \left(-9\right)-3\times 24\\ -4\times \left(-9\right)-5\times 24\end{array}\right]\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{42}\left[\begin{array}{l}-54-72\\ 36-120\end{array}\right]\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{42}\left[\begin{array}{l}-126\\ -84\end{array}\right]=\left[\begin{array}{l}-3\\ -2\end{array}\right]\end{array}$

Hence, from matrix rule, the solution of the given system of equation is $x=-3,y=-2$ .