📘 Question
Solve the matrix equation:
\[
\begin{bmatrix}
x + y \\
x – y
\end{bmatrix}
=
\begin{bmatrix}
2 & 1 \\
4 & 3
\end{bmatrix}
\begin{bmatrix}
1 \\
-2
\end{bmatrix}
\]
Find the value of \((x, y)\).
✏️ Step-by-Step Solution
Step 1: Multiply the matrices
\[
\begin{bmatrix}
2 & 1 \\
4 & 3
\end{bmatrix}
\begin{bmatrix}
1 \\
-2
\end{bmatrix}
=
\begin{bmatrix}
2(1) + 1(-2) \\
4(1) + 3(-2)
\end{bmatrix}
\]
\[
=
\begin{bmatrix}
2 – 2 \\
4 – 6
\end{bmatrix}
=
\begin{bmatrix}
0 \\
-2
\end{bmatrix}
\]
Step 2: Compare both sides
\[
\begin{bmatrix}
x + y \\
x – y
\end{bmatrix}
=
\begin{bmatrix}
0 \\
-2
\end{bmatrix}
\]
So, we get:
- \(x + y = 0\)
- \(x – y = -2\)
Step 3: Solve equations
Add both equations:
\[
2x = -2 \Rightarrow x = -1
\]
Substitute into \(x + y = 0\):
\[
-1 + y = 0 \Rightarrow y = 1
\]
✅ Final Answer
\[
\boxed{(x, y) = (-1, 1)}
\]
💡 Key Concept
Matrix multiplication converts the problem into linear equations. Then solve using basic algebra methods like substitution or elimination.