To solve the system of linear equations:
[tex]\[
\begin{cases}
3x + 4y = 17 \\
-4x - 3y = -18
\end{cases}
\][/tex]
we can use the method of determinants by setting it up as a system of equations in matrix form:
[tex]\[
[A] \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}
\][/tex]
where \([A]\) is the coefficient matrix and \(\begin{bmatrix} c_1 \\ c_2 \end{bmatrix}\) is the constants matrix. Here:
[tex]\[
[A] = \begin{bmatrix} 3 & 4 \\ -4 & -3 \end{bmatrix}, \quad \begin{bmatrix} x \\ y \end{bmatrix}, \quad \text{and} \quad \begin{bmatrix} 17 \\ -18 \end{bmatrix}
\][/tex]
First, we calculate the determinant of the coefficient matrix (\([A]\)):
[tex]\[
\text{det}[A] = (3 \times -3) - (4 \times -4) = -9 - (-16) = -9 + 16 = 7
\][/tex]
Next, we find the determinants for \(x\) and \(y\):
1. The determinant for \(x\):
[tex]\[
\text{det}_{x} = \begin{vmatrix} 17 & 4 \\ -18 & -3 \end{vmatrix} = (17 \times -3) - (4 \times -18) = -51 - 72 = -123
\][/tex]
2. The determinant for \(y\):
[tex]\[
\text{det}_{y} = \begin{vmatrix} 3 & 17 \\ -4 & -18 \end{vmatrix} = (3 \times -18) - (17 \times -4) = -54 + 68 = 14
\][/tex]
Now, we can solve for \(x\) and \(y\):
[tex]\[
x = \frac{\text{det}_{x}}{\text{det}[A]} = \frac{-123}{7} = -17.5714
\][/tex]
[tex]\[
y = \frac{\text{det}_{y}}{\text{det}[A]} = \frac{14}{7} = 2
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\left(-17.5714, 2\right)
\][/tex]
