Question

If $2\left[\begin{array}{cc}3& 4\\ 5& x\end{array}\right]+\left[\begin{array}{cc}1& y\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}7& 0\\ 10& 5\end{array}\right]$, find x − y.

Answer 1

$$\begin{gathered} 2\left[\begin{array}{cc}3& 4\\ 5& x\end{array}\right]+\left[\begin{array}{cc}1& y\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}7& 0\\ 10& 5\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}6+1& 8+y\\ 10+0& 2x+1\end{array}\right]=\left[\begin{array}{cc}7& 0\\ 10& 5\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}7& 8+y\\ 10& 2x+1\end{array}\right]=\left[\begin{array}{cc}7& 0\\ 10& 5\end{array}\right] \\ \Rightarrow 8+y=0\mathrm{and}2x+1=5 \\ \Rightarrow y=-8\mathrm{and}2x=4 \\ \Rightarrow y=-8\mathrm{and}x=2 \\ \mathrm{Hence},x-y=2-(-8)=10. \end{gathered}$$