To solve this system of linear equations:
[tex]\[
\left\{
\begin{array}{l}
6.87x + 7.66y = -9.24 \\
-1.34x + 5.78y = 15.49
\end{array}
\right.
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously.
Start by placing the equations in a matrix form, where [tex]\( A \)[/tex] is the matrix of coefficients and [tex]\( B \)[/tex] is the constant vector:
Matrix A:
[tex]\[
A = \begin{bmatrix}
6.87 & 7.66 \\
-1.34 & 5.78
\end{bmatrix}
\][/tex]
Vector B:
[tex]\[
B = \begin{bmatrix}
-9.24 \\
15.49
\end{bmatrix}
\][/tex]
Using linear algebra techniques (e.g., Gaussian elimination or matrix inversion), solve the system [tex]\( AX = B \)[/tex] for the variable vector [tex]\( X \)[/tex]:
[tex]\[
X = \begin{bmatrix}
x \\
y
\end{bmatrix}
\][/tex]
After solving the system, we find:
[tex]\[
x \approx -3.44, \quad y \approx 1.89
\][/tex]
Thus, the solution to the system of equations is [tex]\( x \approx -3.44 \)[/tex] and [tex]\( y \approx 1.89 \)[/tex].
Given multiple-choice options:
[tex]\[
(1.03, 0.59), \quad (-3.44, 1.89), \quad (-3.75, 1.56), \quad (2.20, -1.32)
\][/tex]
The correct solution is:
[tex]\[
(-3.44, 1.89)
\][/tex]
