Question

Solve the following equations for $X$ and $Y$

$2X-Y=\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right]$ and

$2Y+X=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$

Answer 1

Given:$2X-Y=\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right]$

On multiplying both sides by $2,$ we get

$4X-2Y=\left[\begin{array}{ccc}6& -6& 0\\ 6& 6& 4\end{array}\right]$ .......$\left(1\right)$

Given: $2Y+X=X+2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$ .......$\left(2\right)$

Adding $\left(1\right)$ and $\left(2\right)$ we get

$4X-2Y+X+2Y=\left[\begin{array}{ccc}6& -6& 0\\ 6& 6& 4\end{array}\right]+\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$

$\Rightarrow 5X=\left[\begin{array}{ccc}6+4& -6+1& 0+5\\ 6-1& 6+4& 4-4\end{array}\right]$

$\Rightarrow 5X=\left[\begin{array}{ccc}10& -5& 5\\ 5& 10& 0\end{array}\right]$

$\Rightarrow X=\frac{1}{5}\left[\begin{array}{ccc}10& -5& 5\\ 5& 10& 0\end{array}\right]$

$\therefore X=\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]$

Substitute $X=\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]$ in $\left(2\right)$ we get

$X+2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$

$\Rightarrow 2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]-X$

$\Rightarrow 2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]-\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]$

$\Rightarrow 2Y=\left[\begin{array}{ccc}4-2& 1+1& 5-1\\ -1-1& 4-2& -4-0\end{array}\right]$

$\Rightarrow 2Y=\left[\begin{array}{ccc}2& 2& 4\\ -2& 2& -4\end{array}\right]$

$\Rightarrow Y=\frac{1}{2}\left[\begin{array}{ccc}2& 2& 4\\ -2& 2& -4\end{array}\right]$

$\therefore Y=\left[\begin{array}{ccc}1& 1& 2\\ -1& 1& -2\end{array}\right]$