#### Provide Solution For  R.D. Sharma Maths Class 12 Chapter 5 determinants Exercise 5.4 Question 9 Maths Textbook Solution.

Answer:$\mathrm{x}=\frac{-10}{37} \text { and } \mathrm{y}=\frac{92}{37}$

Hint: Use Cramer’s rule to solve a system of two equations in two variables.

Given:

\begin{aligned} &9 x+5 y=10 \\ &3 y-2 x=8 \Rightarrow-2 x+3 y=8 \end{aligned}

Solution:

First D: determinant of the coefficient matrix

$\mathrm{D}=\left|\begin{array}{cc} 9 & 5 \\ -2 & 3 \end{array}\right|$                                                                $\because\left|\begin{array}{ll} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array}\right|=\left(a_{1} b_{2}-a_{2} b_{1}\right)$

\begin{aligned} &=(9)(3)-(5)(-2) \\ &=27+10 \\ &=37 \end{aligned}

Now, $D\neq 0$. If we are solving for x, the x column is replaced with constant column i.e.

\begin{aligned} \mathrm{D}_{1} &=\left|\begin{array}{cc} 10 & 5 \\ 8 & 3 \end{array}\right| \\ &=30-40 \\ &=-10 \end{aligned}

If we are solving for y, the y column is replaced with constant column i.e.

\begin{aligned} &\mathrm{D}_{2}=\left|\begin{array}{cc} 9 & 10 \\ -2 & 8 \end{array}\right| \\ &=72+20 \\ &=92 \\ &\text { Now, } \mathrm{x}=\frac{D_{1}}{D}=\frac{-10}{37} \\ &\mathrm{y}=\frac{D_{2}}{D}=\frac{92}{37} \end{aligned}

Hence

$y=92 / 37 \text { and } x=-10 / 37$

Concept: Cramer’s rule for system of two equations.

Note: Cramer’s rule will give us unique solution to a system of equations, if it exists. However, if the system has no solution or an infinitive number of solutions that is determinant is zero.