Solve the following system of equations. Express your answer as an ordered pair in the format [tex]$(a,b)$[/tex], with no spaces between the numbers or symbols.

[tex]\[
\begin{array}{l}
3x + 4y = 17 \\
-4x - 3y = -18
\end{array}
\][/tex]

Answer: ________



Answer :

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]