Find x, y, a and b if

Solution:

Since the corresponding elements of two equal matrices are equal,

$\left[\begin{array}{ccc}2 x-3 y & a-b & 3 \\ 1 & x+4 y & 3 a+4 b\end{array}\right]=\left[\begin{array}{ccc}1 & -2 & 3 \\ 1 & 6 & 29\end{array}\right]$

$\Rightarrow 2 x-3 y=1$         ....(1)

$x+4 y=6$

$\Rightarrow x=6-4 y$          .....(2)

Putting the value of $x$ in eq. (1), we get

$2(6-4 y)-3 y=1$

$\Rightarrow 12-8 y-3 y=1$

$\Rightarrow 12-11 y=1$

$\Rightarrow-11 y=-11$

$\Rightarrow y=\frac{-11}{-11}=1$

Putting the value of $y$ in eq. (2), we get

$x=6-4(1)$

$\Rightarrow x=6-4$

$\Rightarrow x=2$

Now,

$a-b=-2$

$\Rightarrow a=-2+b$      ....(3)

$3 a+4 b=29$                  .....(4)

Putting the value of $a$ in eq. (4), we get

$3(-2+b)+4 b=29$

$\Rightarrow-6+3 b+4 b=29$

$\Rightarrow-6+7 b=29$

$\Rightarrow 7 b=29+6$

$\Rightarrow 7 b=35$

$\Rightarrow b=\frac{35}{7}=5$

Putting the value of $b$ in eq. (3), we get

$a=-2+5$

$\Rightarrow a=3$

$\therefore a=3, b=5, x=2$ and $y=1$