#### Please Solve R D Sharma class 12 Chapter 5 determinants Exercise  5.4 Question 3 Maths textbook Solution.

Answer: $\mathrm{x}=7 \text { and } \mathrm{y}=-3$

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

Given: \begin{aligned} &2 x-y=17 \\ &3 x+5 y=6 \end{aligned}

Solution:

First D: determinant of the coefficient matrix

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

\begin{aligned} &=(2)(5)-(3)(-1) \\ &=10+3 \\ &=13 \end{aligned}

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

$\mathrm{D}_{1}=\left|\begin{array}{cc} 17 & -1 \\ 6 & 5 \end{array}\right|$

\begin{aligned} &=(2)(6)-(17)(3) \\ &=12-51 \\ &=-39 \\ &\text { Now, } x=\frac{D_{1}}{D}=\frac{91}{13}=7 \end{aligned}

$\mathrm{y}=\frac{D_{2}}{D}=\frac{-39}{13}=-3$

Hence, x = 7 and y=-3

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.