Question

Find x, y, a and b if

Answer 1

Since the corresponding elements of two equal matrices are equal,

$$\begin{gathered} \left[\begin{array}{ccc}3x+4y& 2& x-2y\\ a+b& 2a-b& -1\end{array}\right]=\left[\begin{array}{ccc}2& 2& 4\\ 5& -5& -1\end{array}\right] \\ \Rightarrow 3x+4y=2...\left(1\right) \\ \Rightarrow x-2y=4 \\ \Rightarrow x=4+2y...\left(2\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ 3\left(4+2y\right)+4y=2 \\ \Rightarrow 12+6y+4y=2 \\ \Rightarrow 12+10y=2 \\ \Rightarrow 10y=2-12 \\ \Rightarrow 10y=-10 \\ \Rightarrow y=\frac{-10}{10}=-1 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ x=4+2\left(-1\right) \\ \Rightarrow x=4-2=2 \\ a+b=5 \\ \Rightarrow a=5-b...\left(3\right) \\ \Rightarrow 2a-b=-5...\left(4\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}a\mathrm{in}\mathrm{eq}.\left(4\right),\mathrm{we}\mathrm{get} \\ 2\left(5-b\right)-b=-5 \\ \Rightarrow 10-2b-b=-5 \\ \Rightarrow 10-3b=-5 \\ \Rightarrow -3b=-15 \\ \Rightarrow b=\frac{-15}{-3} \\ \Rightarrow b=5 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}b\mathrm{in}\mathrm{eq}.\left(3\right),\mathrm{we}\mathrm{get} \\ a=5-5 \\ \Rightarrow a=0 \end{gathered}$$

$\therefore x=2,y=-1,a=0\mathrm{and}b=5$

[3x+4ya+b22a−bx−2y−1]=[25 2−5 4−1][3x+4ya+b22a−bx−2y−1]=[25 2−5 4−1]