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: Comparing the (1,1) entries:
\(2x+3=7\Rightarrow 2x=4\Rightarrow x=2\).

Step 2: Comparing the (2,2) entries:
\(2(y-3)+2=10\Rightarrow 2y-6+2=10\Rightarrow 2y=14\Rightarrow y=7\).

Step 3: Verifying with the other entries:
(1,2): \(2(5)+(-4)=6\) ✓. (2,1): \(2(7)+1=15\) ✓ — both match, confirming consistency.

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