Question

If \(2\begin{bmatrix}x&5\\7&y-3\end{bmatrix}+\begin{bmatrix}3&-4\\1&2\end{bmatrix}=\begin{bmatrix}7&6\\15&10\end{bmatrix}\), find the values of \(x\) and \(y\).

Hint:

Equate corresponding matrix entries after doubling and adding.

Answer 1

Step 1: Isolating the unknown matrix:
\(2\begin{bmatrix}x&5\\7&y-3\end{bmatrix}=\begin{bmatrix}7&6\\15&10\end{bmatrix}-\begin{bmatrix}3&-4\\1&2\end{bmatrix}=\begin{bmatrix}4&10\\14&8\end{bmatrix}\).

Step 2: Dividing by 2:
\(\begin{bmatrix}x&5\\7&y-3\end{bmatrix}=\begin{bmatrix}2&5\\7&4\end{bmatrix}\).

Step 3: Reading off x and y:
Matching entries: \(x=2\) and \(y-3=4\Rightarrow y=7\).

Final Answer:
\[ \boxed{x=2,\ y=7} \]