Chapter 3.4, Problem 51E

To determine

To find: the solutions for the system of equations by using matrix equation.

Answer to Problem 51E

  $x=-1,\text{and }y=7$

Explanation of Solution

Given information:

The given system of equations is as follows:

  $4x+2y=10$..........(1)

  $\begin{array}{l}y=6-x\\ x+y=6\left[\text{Rewritten the equation}\right]\end{array}$..........(2)

Calculation:

The coefficient, variable and constants for the given equations are as follows:

Equation	Coefficient (A)		Variable (X)		Constant (C)
	First	Second	First	Second
$4x+2y=10$	4	2	$x$	$y$	10
$x+y=6$	1	1	$x$	$y$	6

Create a matrix equation from the system of equations as follows:

  $\begin{array}{l}\begin{array}{ccc}A\cdot & X\text{ =}& \text{C}\end{array}\\ \left[\begin{array}{cc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}2\\ 1\end{array}\end{array}\right]\cdot \left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}10\\ 6\end{array}\right]\end{array}$

Evaluate $A^{-1}$ as follows:

  $\begin{array}{l}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}\begin{array}{l}d\\ -c\end{array}& \begin{array}{l}-b\\ a\end{array}\end{array}\right]\\ =\frac{1}{4-2}\left[\begin{array}{cc}\begin{array}{l}1\\ -1\end{array}& \begin{array}{l}-2\\ 4\end{array}\end{array}\right]\\ =\frac{1}{2}\left[\begin{array}{cc}\begin{array}{l}1\\ -1\end{array}& \begin{array}{l}-2\\ 4\end{array}\end{array}\right]\end{array}$

The value of variables can be evaluated as follows:

  $\begin{array}{l}X=A^{-1}C\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}\begin{array}{l}1\\ -1\end{array}& \begin{array}{l}-2\\ 4\end{array}\end{array}\right]\left[\begin{array}{l}10\\ 6\end{array}\right]\left[\text{Substitute values of }{A}^{-1}\text{and C}\right]\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{cc}\begin{array}{l}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}& \begin{array}{l}-1\\ \text{ 2}\end{array}\end{array}\right]\left[\begin{array}{l}10\\ 6\end{array}\right]\\ \left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{c}-1\\ 7\end{array}\right]\end{array}$

Thus, the solution $\left(x,y\right)$ is $\left(-1,7\right)$ .