Solve using matrices.

[tex]\[
\begin{cases}
-13x - 3y = 42 \\
4x + 5y = -23
\end{cases}
\][/tex]

Enter your answer as a decimal and round to the nearest thousandth.



Answer :

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]