Sure, let's solve the matrix equation:
[tex]\[
\left[\begin{array}{cc}-1 & 5 \\ -2 & 8\end{array}\right] = \left[\begin{array}{cc}x + y & x \\ -2 & 8\end{array}\right]
\][/tex]
To find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we need to compare the corresponding elements from both matrices.
1. For the element in the first row and first column:
[tex]\[
-1 = x + y
\][/tex]
2. For the element in the first row and second column:
[tex]\[
5 = x
\][/tex]
3. For the element in the second row and first column:
[tex]\[
-2 = -2
\][/tex]
This equation is already consistent, so we don't need to solve it.
4. For the element in the second row and second column:
[tex]\[
8 = 8
\][/tex]
This equation is also already consistent, so we don't need to solve it.
Now, let's solve the system of equations:
From equation (2):
[tex]\[
5 = x
\][/tex]
We have directly that:
[tex]\[
x = 5
\][/tex]
Next, substitute the value of [tex]\(x\)[/tex] into equation (1):
[tex]\[
-1 = x + y
\][/tex]
Substitute [tex]\(x = 5\)[/tex]:
[tex]\[
-1 = 5 + y
\][/tex]
Now, solve for [tex]\(y\)[/tex]:
[tex]\[
y = -1 - 5
\][/tex]
[tex]\[
y = -6
\][/tex]
Therefore, the solution is:
[tex]\[
x = 5 \quad \text{and} \quad y = -6
\][/tex]
Thus, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the given matrix equation are:
[tex]\[
(x, y) = (5, -6)
\][/tex]
