#### Need Solution for R.D.Sharma Maths Class 12 Chapter 5 determinants  Exercise 5.4 Question 22  Maths Textbook Solution.

Answer:$\mathrm{x}=\infty \text { and } \mathrm{y}=\infty$

Hint: Solving determinant gives zero.

Given:

\begin{aligned} &2 x-y=5 \\ &4 x-2 y=7 \end{aligned}

Solution:

$2 x-y=5$                                                                    .....(1)

\begin{aligned} \Rightarrow 4 x-2 y=7 & \Rightarrow 2(2 x-y)=7 \\ & \Rightarrow 2 x-y=\frac{7}{2} \end{aligned}                           .....(2)

Now, different value of 2x – y is not possible. So, the linear equations are inconsistent.

Solving determinant,

$|\mathrm{A}|=\left|\begin{array}{ll} 2 & -1 \\ 4 & -2 \end{array}\right| \quad \because\left|\begin{array}{ll} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array}\right|=\left(a_{1} b_{2}-a_{2} b_{1}\right)$

\begin{aligned} &=-4+4 \\ &\mathrm{D}=0 \end{aligned}

\begin{aligned} &\mathrm{D}_{\mathrm{x}}=\left|\begin{array}{ll} 5 & -1 \\ 7 & -2 \end{array}\right|=-10+7=-3 \neq 0 \\ &\mathrm{D}_{\mathrm{y}}=\left|\begin{array}{ll} 2 & 5 \\ 4 & 7 \end{array}\right|=14-20=-6 \neq 0 \end{aligned}

By Cramer’s rule,

\begin{aligned} &\Rightarrow x=\frac{D_{x}}{D}=\frac{-3}{0}=\infty\\ &\Rightarrow y=\frac{D_{y}}{D}=\frac{-6}{0}=\infty\\ &\text { Since, } \mathrm{D}=0 \text { and } \mathrm{D}_{x} \text { and } \mathrm{D}_{\mathrm{y}} \neq 0\\ &\therefore \text { Linear equations are inconsistent. } \end{aligned}

Concept: Solving matrix of order 2x2 by solving linear equations

Note: When D = 0, there is either no solution or infinite solutions.