# Find the values of a, b, c and d from the following equations:

Question:

Find the values of abc and d from the following equations:

$\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]$

Solution:

Since all the corresponding elements of a matrix are equal,

$\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]$

$\Rightarrow 2 a+b=4$

$\Rightarrow b=4-2 a$           ....(1)

$a-2 b=-3$                            ....(2)

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

$a-2(4-2 a)=-3$

$\Rightarrow a-8+4 a=-3$

$\Rightarrow 5 a-8=-3$

$\Rightarrow 5 a=-3+8$

$\Rightarrow 5 a=5$

$\Rightarrow a=1$

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

$b=4-2(1)$

$\Rightarrow b=4-2$

$\Rightarrow b=2$

$5 c-d=11$

$\Rightarrow 5 c-11=d$        .....(3)

$4 c+3 d=24$                       .....(4)

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

$4 c+3(5 c-11)=24$

$\Rightarrow 4 c+15 c-33=24$

$\Rightarrow 19 c-33=24$

$\Rightarrow 19 c=24+33$

$\Rightarrow 19 c=57$

$\Rightarrow c=\frac{57}{19}=3$

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

$5(3)-11=d$

$\Rightarrow 15-11=d$

$\Rightarrow d=4$

$\therefore a=1, b=2, c=3$ and $d=4$