To solve the matrix equation for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], follow these steps:
The given matrix equation is:
[tex]\[
\frac{1}{2} \left[\begin{array}{cc}
4 & 8 \\
x+3 & -4
\end{array}\right] - 3 \left[\begin{array}{cc}
1 & y+1 \\
-1 & -2
\end{array}\right] = \left[\begin{array}{cc}
-1 & -5 \\
7 & 4
\end{array}\right]
\][/tex]
First, simplify the left side of the equation.
Calculate the scaled matrices:
[tex]\[
\frac{1}{2} \left[\begin{array}{cc}
4 & 8 \\
x+3 & -4
\end{array}\right] = \left[\begin{array}{cc}
\frac{4}{2} & \frac{8}{2} \\
\frac{x+3}{2} & \frac{-4}{2}
\end{array}\right] = \left[\begin{array}{cc}
2 & 4 \\
\frac{x+3}{2} & -2
\end{array}\right]
\][/tex]
[tex]\[
3 \left[\begin{array}{cc}
1 & y+1 \\
-1 & -2
\end{array}\right] = \left[\begin{array}{cc}
3 \cdot 1 & 3(y+1) \\
3(-1) & 3(-2)
\end{array}\right] = \left[\begin{array}{cc}
3 & 3y+3 \\
-3 & -6
\end{array}\right]
\][/tex]
Now, subtract the two matrices:
[tex]\[
\left[\begin{array}{cc}
2 & 4 \\
\frac{x+3}{2} & -2
\end{array}\right] - \left[\begin{array}{cc}
3 & 3y+3 \\
-3 & -6
\end{array}\right] = \left[\begin{array}{cc}
2 - 3 & 4 - (3y + 3) \\
\frac{x+3}{2} + 3 & -2 - (-6)
\end{array}\right]
= \left[\begin{array}{cc}
-1 & 1 - 3y \\
\frac{x+3}{2} + 3 & 4
\end{array}\right]
\][/tex]
Set this result equal to the given matrix:
[tex]\[
\left[\begin{array}{cc}
-1 & 1 - 3y \\
\frac{x+3}{2} + 3 & 4
\end{array}\right] = \left[\begin{array}{cc}
-1 & -5 \\
7 & 4
\end{array}\right]
\][/tex]
Now, match the corresponding elements:
1. Top-left:
[tex]\[
-1 = -1
\][/tex]
This is true, so it is correct.
2. Top-right:
[tex]\[
1 - 3y = -5
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
1 - 3y = -5 \\
1 + 5 = 3y \\
6 = 3y \\
y = 2
\][/tex]
3. Bottom-left:
[tex]\[
\frac{x+3}{2} + 3 = 7
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{x+3}{2} + 3 = 7 \\
\frac{x+3}{2} = 4 \\
x + 3 = 8 \\
x = 5
\][/tex]
4. Bottom-right:
[tex]\[
4 = 4
\][/tex]
This is true, so it is correct.
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[
\boxed{x = 5, y = 2}
\][/tex]
