Question

Find the value of $X$ and $Y$ if
$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$ and $3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$

Answer 1

Find the value of $X$ and $Y$ if

$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$ $3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$

$\to$ let Assume $\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$ = A

$\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$ = B

$\to 2x+3y=A$ __ (i)

$3x+2y=B$ __ (ii)

$\to$ multiple equation (i) with 2 & equation (ii) with (3)

$4x+6y=2A$

$9x+6y=3B$

________________

$-5x=2A-3B$

$\therefore x=\frac{-1}{5}[2A-3B]$

$\therefore x=\frac{3B}{5}-\frac{2A}{5}$

$=\frac{3}{5}\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]-\frac{2}{5}\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$

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

$=\left[\begin{array}{cccc}6/5& -4/5& -4/5& -6/5\\ -3/5& -8/5& 3& -0\end{array}\right]$

$\therefore x=\left[\begin{array}{cc}1/5& -12/5\\ -11/5& 3\end{array}\right]$

$\to$ value of x put in equation (i)

$\therefore \left[\begin{array}{cc}2/5& -24/5\\ -22/5& 6\end{array}\right]+3y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$

$\therefore 3y=\left[\begin{array}{cc}2-2/5& 3+24/5\\ 4+22/5& 0-6\end{array}\right]=\left[\begin{array}{cc}8/5& 39/5\\ 42/5& -6\end{array}\right]$

$\therefore y=\left[\begin{array}{cc}8/15& 39/15\\ 42/15& -2\end{array}\right]$