#### Need Solution for R.D.Sharma Maths Class 12 Chapter 5 determinants  Exercise 5.4 Question 16 Maths Textbook Solution.

Answer:$x=1, y=-1 \text { and } z=-1$

Hint: Use Cramer’s rule to solve a system of linear equations

Given:

\begin{aligned} &5 x-7 y+z=11 \\ &6 x-8 y-z=15 \\ &3 x+2 y-6 z=7 \end{aligned}

Solution:

First take coefficient of variables x, y and z.

$|\mathrm{A}|=\left|\begin{array}{ccc} 5 & -7 & 1 \\ 6 & -8 & -1 \\ 3 & 2 & -6 \end{array}\right|$                                                                        $\because$(Taking first row for solving determinant)

\begin{aligned} &=5(48+2)+7(-36+3)+1(12+24) \\ &=5(50)+7(-33)+1(36) \\ &=250-231+36 \\ &=55 \end{aligned}

Now for x, the x column is replaced with constant column i.e.

\begin{aligned} \mathrm{D}_{\mathrm{x}} &=\left|\begin{array}{ccc} 11 & -7 & 1 \\ 15 & -8 & -1 \\ 7 & 2 & -6 \end{array}\right| \\ &=11(48+2)+7(-90+7)+1(30+56) \\ &=11(50)+7(-83)+86 \\ &=550-581+86 \\ &=55 \end{aligned}

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

\begin{aligned} \mathrm{D}_{\mathrm{y}} &=\left|\begin{array}{ccc} 5 & 11 & 1 \\ 6 & 15 & -1 \\ 3 & 7 & -6 \end{array}\right| \\ &=5(-90+7)-11(-36+3)+1(42-45) \\ &=5(-83)-11(-33)+1(-3) \end{aligned}

\begin{aligned} &=-415+363-3 \\ &=-55 \end{aligned}

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

\begin{aligned} \mathrm{D}_{z} &=\left|\begin{array}{ccc} 5 & -7 & 11 \\ 6 & -8 & 15 \\ 3 & 2 & 7 \end{array}\right| \\ &=5(-56-30)+7(42-45)+11(12+24) \\ &=5(-86)+7(-3)+11(36) \\ &=-430-21+396 \\ &=-55 \end{aligned}

By Cramer’s rule,

\begin{aligned} &\Rightarrow x=\frac{D_{x}}{D}=\frac{55}{55}=1 \\ &\Rightarrow y=\frac{D_{y}}{D}=\frac{-55}{55}=-1 \\ &\Rightarrow z=\frac{D_{z}}{D}=\frac{-55}{55}=-1 \end{aligned}

Concept: Determinant solving of 3 x 3 matrix (Cramer’s rule)

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.