Get Answers to all your Questions

header-bg qa

Explain Solution R.D.Sharma Class 12 Chapter 5 deteminants Exercise 5.4 Question 13 maths Textbook Solution.

Answers (1)

Answer: \mathrm{x}=1, \mathrm{y}=2 \text { and } \mathrm{z}=1

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

Given:

\begin{aligned} &6 x+y-3 z=5 \\ &x+3 y-2 z=5 \\ &2 x+y+4 z=8 \end{aligned}

Solution:

First take coefficient of variables x, y and z.

|\mathrm{A}|=\left|\begin{array}{ccc} 6 & 1 & -3 \\ 1 & 3 & -2 \\ 2 & 1 & 4 \end{array}\right|                                                            \because(Taking first row for solving determinant)

\begin{aligned} &=6(12+2)-1(4+4)-3(1-6) \\ &=6(14)-1(8)-3(-5) \\ &=84-8+15 \\ &=91 \end{aligned}

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

\mathrm{D}_{\bar{x}}=\left|\begin{array}{ccc} 5 & 1 & -3 \\ 5 & 3 & -2 \\ 8 & 1 & 4 \end{array}\right|

\begin{aligned} &=5(12+2)-1(20+16)-3(5-24) \\ &=5(14)-1(36)-3(-19) \\ &=70-36+57 \\ &=91 \end{aligned}

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

\mathrm{D}_{\mathrm{x}}=\left|\begin{array}{ccc} 6 & 5 & -3 \\ 1 & 5 & -2 \\ 2 & 8 & 4 \end{array}\right|

\begin{aligned} &=6(20+16)-5(4+4)-3(8+10) \\ &=6(36)+3(-2)-5(8) \\ &=216-6-40 \\ &=182 \end{aligned}

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

\begin{aligned} &D_{z}=\left|\begin{array}{lll} 6 & 1 & 5 \\ 1 & 3 & 5 \\ 2 & 1 & 8 \end{array}\right| \\ &=6(24-5)-1(8-10)+5(1-6) \\ &=6(19)-1(-2)+5(-5) \\ &=114+2-25 \\ &=91 \end{aligned}

By Cramer’s rule,

\begin{aligned} &\Rightarrow x=\frac{D_{x}}{D}=\frac{91}{91}=1 \\ &\Rightarrow y=\frac{D_{y}}{D}=\frac{182}{91}=2 \\ &\Rightarrow z=\frac{D_{z}}{D}=\frac{91}{91}=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.

 

Posted by

infoexpert21

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads