Answer
$x_{1}$ = 1
$x_{2}$ = 7
Work Step by Step
Cramer's Rule is used to find the solutions to a system of equations.
We first have to turn the system of equations into matrices.
A = $\begin{bmatrix}
-5 & 2 \\
3 & -1
\end{bmatrix}$
We then replace the first column with the column of solutions.
$A_{1}$(b) = $\begin{bmatrix}
9 & 2 \\
-4 & -1
\end{bmatrix}$
We then replace the second column with the column of solutions.
$A_{2}$(b) = $\begin{bmatrix}
-5 & 9 \\
3 & -4
\end{bmatrix}$
We then need to find the determinant of each matrix.
Note: The determinant of a 2x2 matrix is ad-bc where $\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$
det(A) = (-5)(-1) - (2)(3) = 5 - 6 = -1
det($A_{1}$(b)) = (9)(-1) - (-4)(2) = -9 + 8 = -1
det($A_{2}$(b)) = (-5)(-4) - (9)(3) = 20 - 27 = -7
We then use the formula:
$x_{1}$ = $\frac{det(A_{1}(b))}{det(A)}$ = $\frac{-1}{-1}$ = 1
$x_{2}$ = $\frac{det(A_{2}(b))}{det(A)}$ = $\frac{-7}{-1}$ = 7
The solutions to the system are:
$x_{1}$ = 1
$x_{2}$ = 7
