Question

If $x\left[\begin{array}{c}2\\ 3\end{array}\right]+y\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$, find the value of x.

Answer 1

$$\begin{gathered} x\left[\begin{array}{c}2\\ 3\end{array}\right]+y\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}2x-y\\ 3x+y\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right] \\ \mathrm{Corresponding}\mathrm{elements}\mathrm{of}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \Rightarrow 2x-y=10\mathrm{and}3x+y=5 \\ \Rightarrow y=2x-10\mathrm{and}3x+\left(2x-10\right)=5 \\ \Rightarrow y=2x-10\mathrm{and}5x=15 \\ \Rightarrow y=2x-10\mathrm{and}x=3 \\ \Rightarrow y=2\left(3\right)-10\mathrm{and}x=3 \\ \Rightarrow y=-4\mathrm{and}x=3 \\ \therefore x=3\mathrm{and}y=-4. \end{gathered}$$

Hence, the value of x is 3.