📘 Question
Solve the matrix equation:
\[
2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix}
+
\begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix}
=
\begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}
\]
Find the value of \(x – y\).
✏️ Step-by-Step Solution
Step 1: Multiply the matrix by 2
\[
2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix}
=
\begin{bmatrix}6 & 8 \\ 10 & 2x\end{bmatrix}
\]
Step 2: Add the matrices
\[
\begin{bmatrix}6 & 8 \\ 10 & 2x\end{bmatrix}
+
\begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix}
=
\begin{bmatrix}7 & 8 + y \\ 10 & 2x + 1\end{bmatrix}
\]
Step 3: Compare corresponding elements
\[
\begin{bmatrix}7 & 8 + y \\ 10 & 2x + 1\end{bmatrix}
=
\begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}
\]
Equating elements:
- \(8 + y = 0 \Rightarrow y = -8\)
- \(2x + 1 = 5 \Rightarrow 2x = 4 \Rightarrow x = 2\)
Step 4: Find \(x – y\)
\[
x – y = 2 – (-8)
\]
\[
= 10
\]
✅ Final Answer
\[
\boxed{10}
\]
💡 Key Concept
In matrix equations, corresponding elements of equal matrices are always equal. This allows us to form simple equations and solve unknown variables easily.