To solve the given system of linear equations using matrices and row operations, we can proceed by converting the system into an augmented matrix and then use Gaussian elimination to find the solution. Let's consider the following system of equations:
[tex]\[
\left\{
\begin{array}{l}
2x - 8y = -6 \\
3x + 3y = 2
\end{array}
\right.
\][/tex]
Step 1: Write the augmented matrix corresponding to the system of equations.
[tex]\[
\left[\begin{array}{cc|c}
2 & -8 & -6 \\
3 & 3 & 2
\end{array}\right]
\][/tex]
Step 2: Perform row operations to row reduce the augmented matrix to row echelon form.
Let's start by using the first row to eliminate the [tex]\(x\)[/tex]-term in the second row. First, multiply the first row by [tex]\(\frac{3}{2}\)[/tex] to make the coefficient of [tex]\(x\)[/tex] in the first row 3:
[tex]\[
\left[\begin{array}{cc|c}
3 & -12 & -9 \\
3 & 3 & 2
\end{array}\right]
\][/tex]
Next, subtract the first row from the second row to eliminate the [tex]\(x\)[/tex]-term in the second row:
[tex]\[
\left[\begin{array}{cc|c}
3 & -12 & -9 \\
0 & 15 & 11
\end{array}\right]
\][/tex]
Step 3: Normalize the rows.
Divide the second row by 15 to make the leading coefficient in the second row 1:
[tex]\[
\left[\begin{array}{cc|c}
3 & -12 & -9 \\
0 & 1 & \frac{11}{15}
\end{array}\right]
\][/tex]
Now, use the second row to eliminate the [tex]\(y\)[/tex]-term in the first row. Multiply the second row by 12 and add it to the first row:
[tex]\[
\left[\begin{array}{cc|c}
3 & 0 & -9 + 12 \left(\frac{11}{15}\right) \\
0 & 1 & \frac{11}{15}
\end{array}\right]
\][/tex]
Simplifying the first row:
[tex]\[
\left[\begin{array}{cc|c}
3 & 0 & -9 + \frac{132}{15} \\
0 & 1 & \frac{11}{15}
\end{array}\right]
\][/tex]
Calculate:
[tex]\[
-9 + \frac{132}{15} = -9 + 8.8 = -0.2
\][/tex]
Thus, the augmented matrix is:
[tex]\[
\left[\begin{array}{cc|c}
3 & 0 & -0.2 \\
0 & 1 & \frac{11}{15}
\end{array}\right]
\][/tex]
Finally, divide the first row by 3 to normalize it:
[tex]\[
\left[\begin{array}{cc|c}
1 & 0 & -0.0666666666666667 \\
0 & 1 & 0.733333333333333
\end{array}\right]
\][/tex]
From the reduced matrix, we can read off the solutions:
[tex]\[
x = -0.0666666666666667
\][/tex]
[tex]\[
y = 0.733333333333333
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
A. \text{The solution is } \left(x = -0.0666666666666667, \ y = 0.733333333333333 \right).
\][/tex]
