Question

Use a matrix to solve the system. 8. 2x + 6y = 38 (5x - y = 15

Answer 1

Answer:

x=4 and y=5

Step-by-step explanation:

The given system of equations are

$2x+6y=38$

$5x-y=15$

The matrix form is

$\begin{bmatrix}2&6\\ \:5&-1\end{bmatrix}\begin{bmatrix}x\\ \:y\end{bmatrix}=\begin{bmatrix}38\\ \:15\end{bmatrix}$

Let as assume

$A=\begin{bmatrix}2&6\\ \:5&-1\end{bmatrix}$

$X=\begin{bmatrix}x\\ \:y\end{bmatrix}$

$B=\begin{bmatrix}38\\ \:15\end{bmatrix}$

then,

$AX=B$

$X=A^{-1}B$

We know that,

$\begin{bmatrix}a\:&\:b\:\\ c\:&\:d\:\end{bmatrix}^{-1}=\frac{1}{\det \begin{bmatrix}a\:&\:b\:\\ c\:&\:d\:\end{bmatrix}}\begin{bmatrix}d\:&\:-b\:\\ -c\:&\:a\:\end{bmatrix}$

$A^{-1}=\frac{1}{\det \begin{bmatrix}2&6\\ 5&-1\end{bmatrix}}\begin{bmatrix}-1&-6\\ -5&2\end{bmatrix}$

$A^{-1}=\frac{1}{-32}\begin{bmatrix}-1&-6\\ -5&2\end{bmatrix}$

$X=\frac{1}{-32}\begin{bmatrix}-1&-6\\ -5&2\end{bmatrix}\begin{bmatrix}38\\ \:15\end{bmatrix}$

$X=\frac{1}{-32}\begin{bmatrix}\left(-1\right)\cdot \:38+\left(-6\right)\cdot \:15\\ \left(-5\right)\cdot \:38+2\cdot \:15\end{bmatrix}$

$X=\frac{1}{-32}\begin{bmatrix}-128\\ -160\end{bmatrix}$

$\begin{bmatrix}x\\ \:y\end{bmatrix}=\begin{bmatrix}4\\ 5\end{bmatrix}$

Therefore, the value of x is 4 and value of y is 5.