Chapter 11, Problem 2CRT

a.

To determine

To find: The Solution of the linear system and find the value of x, y and z.

Answer to Problem 2CRT

The solution of the system is

  $x=\frac{-11}{3}$

  $\begin{array}{l}y=\frac{-10}{3}\\ z=\frac{-7}{3}\end{array}$

Explanation of Solution

Given:

The set of equation

  $\begin{array}{l}x+y+z=2\\ 2x+3y-z=5\\ 3x+5y+2z=1\end{array}$

Concept Used:

1. Determinant of any mented matrix is calculated. If determinant is non zero then surface is exist. If determinant value of zero then there is no solution of given set of equation.

2. Elimination method is used to solve the equation.

Calculation:

Writing in the matrix form of given set of equations

  $\left[\begin{array}{ccc}1& 1& -1\\ 2& 3& -1\\ 3& 5& 2\end{array}\right]$

Find the determinant of above matrix.

  $\begin{array}{l}D=1\left[\left(3\times 2-5\times 1\right)-1\left(2\times 2-3\times -1\right)-1\left(2\times 5-3\times 3\right)\right]\\ =\left(6+5\right)-\left(4+3\right)-\left(10-9\right)\\ D=11-8=3\\ \therefore D=3\end{array}$

Since determinant is non − zero hence there are unique solution for given set of equation.

Using elimination method, eliminate z from set of equation

  $\begin{array}{l}x+y-z=2\\ \Rightarrow z=x+y-2\end{array}$

Using z in rest of two equation we get

  $\begin{array}{l}2x+3y-\left(x+y-2\right)=5\\ 3x+5y+2\left(x+y-2\right)=1\end{array}\}$

or $\begin{array}{l}x+2y=3\\ 5x+7y=5\end{array}\}$

Multiplying first equation by $5$ and rewrite the set of equation $$

  $\begin{array}{l}5x+10y=5\\ 5x+7y=5\\ ---\\ \Rightarrow 3y=10\\ \Rightarrow y=\frac{10}{3}\end{array}$

Putting u value in $x+2y=3$

  $\begin{array}{l}\Rightarrow x+2\times \frac{10}{3}=3\\ \Rightarrow x=3-\frac{20}{3}\\ \Rightarrow x=\frac{-11}{3}\end{array}$

Using x and y value in $x+y-z=2$

We get $\frac{-11}{3}+\frac{10}{3}-z=2$

  $\begin{array}{l}\Rightarrow \frac{-1}{3}-z=2\\ \Rightarrow z=\frac{-1}{3}-2\\ \Rightarrow z=\frac{-7}{3}\end{array}$

Hence $x=\frac{-11}{3}$

  $\begin{array}{l}y=\frac{-10}{3}\\ z=\frac{-7}{3}\end{array}$

Conclusion:

The required values of x. y and z are

  $x=\frac{-11}{3}$

  $\begin{array}{l}y=\frac{-10}{3}\\ z=\frac{-7}{3}\end{array}$

b.

To determine

To write: The coefficient of x, y, z in matrix form.

Answer to Problem 2CRT

  $\left[\begin{array}{ccc}0& 1& -1\\ 1& 2& -3\\ 3& 5& -8\end{array}\right]$

Explanation of Solution

Given information:

  $\begin{array}{l}x+y+z=2\\ 2x+3y-z=5\\ 3x+5y+2z=1\end{array}$

Concept Used:

  $\left[\begin{array}{ccc}0& 1& -1\\ 1& 2& -3\\ 3& 5& -8\end{array}\right]$

The determinant of matrix

Calculation:

The determinant of matrix is given below:

  $\begin{array}{l}D=\left[\left(2\times -8\right)-\left(5\times -3\right)\right]-1\left[\left(1\times -8\right)-\left(3\times -3\right)\right]-\left[\left(1\times 5\right)-\left(2\times 3\right)\right]\\ =0+1-1\\ D=0\end{array}$

Conclusion:

Determinant is zero, hence this set of equation has no solution. $$