# Solve the equation for x, y, z and t if

Question:

Solve the equation for $x, y, z$ and $t$ if

$2\left[\begin{array}{ll}x & z \\ y & t\end{array}\right]+3\left[\begin{array}{rr}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$

Solution:

$2\left[\begin{array}{ll}x & z \\ y & t\end{array}\right]+3\left[\begin{array}{rr}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}2 x & 2 z \\ 2 y & 2 t\end{array}\right]+\left[\begin{array}{lr}3 & -3 \\ 0 & 6\end{array}\right]=\left[\begin{array}{ll}9 & 15 \\ 12 & 18\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}2 x+3 & 2 z-3 \\ 2 y & 2 t+6\end{array}\right]=\left[\begin{array}{ll}9 & 15 \\ 12 & 18\end{array}\right]$

Comparing the corresponding elements of these two matrices, we get:

$2 x+3=9$

$\Rightarrow 2 x=6$

$\Rightarrow x=3$

$2 y=12$

$\Rightarrow y=6$

$2 z-3=15$

$\Rightarrow 2 z=18$

$\Rightarrow z=9$

$2 t+6=18$

$\Rightarrow 2 t=12$

$\Rightarrow t=6$

$\therefore x=3, y=6, z=9$, and $t=6$