Question

Find X and Y if $2X+3Y=(\begin{array}{cc}2& 3\\ 4& 0\end{array})$ and $3X+2Y=(\begin{array}{cc}2& -2\\ -1& 5\end{array})$.

Answer 1

$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]⟶\left(1\right)$

$\Rightarrow 3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]⟶\left(2\right)$

Multily equation $\left(1\right)$ by $3$ and $\left(2\right)$ by $2,$ we get

$\Rightarrow 6X+9Y=\left[\begin{array}{cc}6& 9\\ 12& 0\end{array}\right]⟶\left(3\right)$

$\Rightarrow 6X+4Y=\left[\begin{array}{cc}4& -4\\ -2& 10\end{array}\right]⟶\left(4\right)$

Subtracting, we get

$\Rightarrow 5Y=\left[\begin{array}{cc}6& 9\\ 12& 0\end{array}\right]-\left[\begin{array}{cc}4& -4\\ -2& 10\end{array}\right]$

$\Rightarrow 5Y=\left[\begin{array}{cc}2& 13\\ 14& -10\end{array}\right]$

$\Rightarrow Y=\frac{1}{5}\left[\begin{array}{cc}2& 13\\ 14& -10\end{array}\right]$

Now, multiply equation $\left(1\right)$ by $2$ $\&$ $\left(2\right)$ by $3,$ we get

$\Rightarrow 4X+6Y=\left[\begin{array}{cc}4& 6\\ 8& 0\end{array}\right]⟶\left(5\right)$

$\Rightarrow 9X+6Y=\left[\begin{array}{cc}6& -6\\ -3& 15\end{array}\right]⟶\left(6\right)$

Subtracting , we get [ equation $\left(6\right)-$ equation $\left(5\right)$ ]

$\Rightarrow 5X=\left[\begin{array}{cc}6& -6\\ -3& 15\end{array}\right]-\left[\begin{array}{cc}4& 6\\ 8& 0\end{array}\right]$

$\Rightarrow 5X=\left[\begin{array}{cc}2& -12\\ -11& 15\end{array}\right]$

$\Rightarrow X=\frac{1}{5}\left[\begin{array}{cc}2& -12\\ -11& 15\end{array}\right]$

$\Rightarrow Y=\frac{1}{5}\left[\begin{array}{cc}2& 13\\ 14& -10\end{array}\right]$

Hence, the answer is $X=\frac{1}{5}\left[\begin{array}{cc}2& -12\\ -11& 15\end{array}\right],Y=\frac{1}{5}\left[\begin{array}{cc}2& 13\\ 14& -10\end{array}\right].$