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

Answer:$\mathrm{x}=\frac{-5}{11} \text { and } \mathrm{y}=\frac{12}{11}$

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

Given: \begin{aligned} &2 x-y=-2 \\ &3 x+4 y=3 \end{aligned}

Solution:

First D: determinant of the coefficient matrix

$\mathrm{D}=\left|\begin{array}{cc} 2 & -1 \\ 3 & 4 \end{array}\right| \quad \because\left|\begin{array}{ll} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array}\right|=\left(a_{1} b_{2}-a_{2} b_{1}\right)$

\begin{aligned} &=(2)(4)-(3)(-1) \\ &=8+3 \\ &=11 \end{aligned}

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

$\mathrm{D}_{1}=\left|\begin{array}{cc} -2 & -1 \\ 3 & 4 \end{array}\right|$

\begin{aligned} &=(4)(-2)-(3)(-1) \\ &=-8+3 \\ &=-5 \end{aligned}

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

$\mathrm{D}_{2}=\left|\begin{array}{cc} 2 & -2 \\ 3 & 3 \end{array}\right|$

\begin{aligned} &=(2)(3)-(3)(-2) \\ &=6+6 \\ &=12 \end{aligned}

Now, \begin{aligned} &\mathrm{x}=\frac{D_{1}}{D}=\frac{-5}{11} \\ &\mathrm{y}=\frac{D_{2}}{D}=\frac{12}{11} \end{aligned}

Hence,$\mathrm{x}=\frac{-5}{11} \text { and } \mathrm{y}=\frac{12}{11}$

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.