Answer to Question #236240 in Linear Algebra for luka

Question

Using matrix method, solve the simultaneous equations {x-3y=3

5x-9y=11

Answer 1

$A\bar x=\bar b$

$A=\begin{pmatrix} 1 & -3 \\ 5 & -9 \end{pmatrix},\bar b=\begin{pmatrix} 3 \\ 11 \end{pmatrix}$ .

$det(A)=-9+15=6.$

$A^{-1}=\frac{1}{6}\begin{pmatrix} -9 & 3 \\ -5 & 1 \end{pmatrix}$

$\bar x=\begin{pmatrix} x \\ y \end{pmatrix}=A^{-1}\bar b=\frac{1}{6}\begin{pmatrix} -9& 3 \\ -5 & 1 \end{pmatrix}\begin{pmatrix} 3 \\ 11 \end{pmatrix}=\frac{1}{6}\begin{pmatrix} 6 \\ -4 \end{pmatrix}=\begin{pmatrix} 1 \\ -\frac{2}{3} \end{pmatrix}$

Thus, $x=1,y=-\frac{2}{3}.$