Q.Find $x$ and $y$ if $\begin{bmatrix}x+y & 2\\5 & x-y\end{bmatrix}=\begin{bmatrix}6&2\\5&2\end{bmatrix}$.

[!TLDR] Match corresponding entries: $x+y=6$ from position $(1,1)$, $x-y=2$ from position $(2,2)$. [!ANSWER] $x=4,\ y=2$.

By the definition of matrix equality (Section 3), corresponding entries of the two matrices must be equal. Comparing position $(1,1)$: $x+y=6$. Comparing position $(2,2)$: $x-y=2$. (Positions $(1,2)$ and $(2,1)$ already agree, $2=2$ and $5=5$, and give no new information.) Adding the two equations: $(x+y)+(x-y)=6+2 \Rightarrow 2x=8 \Rightarrow x=4$. Substituting back: $4+y=6 \Rightarrow y=2$. [!ANSWER] $x=4,\ y=2$.

Use equality of corresponding entries to write down every independent equation the matrix equality implies, then solve the resulting system simultaneously.

A common mistake is writing an equation for every one of the four positions and then getting confused when two of them are identical ("$2=2$") rather than recognising these carry no information about $x,y$.