To solve the given system of linear equations using matrices, we need to follow these steps:
1. Represent the system of equations in matrix form:
[tex]\[
\begin{cases}
-13x - 3y = 42 \\
4x + 5y = -23
\end{cases}
\][/tex]
2. Write the augmented matrix, which combines the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] with the constants from the right-hand side of the equations:
[tex]\[
\left[ \begin{array}{cc|c}
-13 & -3 & 42 \\
4 & 5 & -23
\end{array} \right]
\][/tex]
3. To find the solution, we can use the inverse of the coefficient matrix. Write the coefficient matrix [tex]\(A\)[/tex] and the constants vector [tex]\(B\)[/tex]:
[tex]\[
A = \left[ \begin{array}{cc}
-13 & -3 \\
4 & 5
\end{array} \right]
\][/tex]
[tex]\[
B = \left[ \begin{array}{cc}
42 \\
-23
\end{array} \right]
\][/tex]
4. Solve for the vector [tex]\(X\)[/tex] by using the inverse of the matrix [tex]\(A\)[/tex]. The solution vector [tex]\(X\)[/tex] satisfies the equation [tex]\(AX = B\)[/tex]:
[tex]\[
X = A^{-1}B
\][/tex]
5. After performing the calculations, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are obtained.
6. The solutions are rounded to the nearest thousandth:
[tex]\[
x \approx -2.660
\][/tex]
[tex]\[
y \approx -2.472
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (-2.660, -2.472)
\][/tex]
