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Please solve rd sharma class 12 chapter Solution of simultaneous linear equation exercise 7.2 question 5 maths textbook solution

Answers (1)

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Answer:

$x= 2k,y= -3k,z= k$

Hint:

If $\left | A \right |= 0$ , then the system of equation has non-trivial solution.

Given:

$x+y+z=0$

$x-y-5z=0$

$x+2y+4z=0$

Explanation:

The given homogeneous system can be written in matrix form

i.e. $\begin{bmatrix} 1 & 1 & 1\\ 1& -1 & -5\\ 1& 2 & 4\end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}=\begin{bmatrix} 0\\ 0\\0 \end{bmatrix}$

$A=\begin{bmatrix} 1 & 1 & 1\\ 1& -1 & -5\\ 1& 2& 4 \end{bmatrix}$

let

$\left | A \right |=\begin{vmatrix} 1 & 1 & 1\\ 1& -1 & -5\\ 1& 2 & 4 \end{vmatrix}$

$= 1(-4+10)-1(4+5)+1(2+1)$

$= 6-9+3$

$= 0$

So, the given system of equation has a non-trivial solution.

To find these solutions, we write the first two equations as

$x+y=6z$

$x-y=-2z$

Let $z=k$ , then we have

$\begin{bmatrix} 1 & 1\\ 1 & -1\end{bmatrix}\begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix} -k\\ 5k\end{bmatrix}$

$A=\begin{bmatrix} 1 & 1\\ 1 & -1\end{bmatrix},X=\begin{bmatrix} x\\ y\end{bmatrix},B=\begin{bmatrix} -k\\ -5k\end{bmatrix}$

Where,

$\left | A \right |=\begin{vmatrix} 1 & 1\\ 1 & -1\end{vmatrix}=-2\neq 0$

So $A^{-1}$ exists

^Now,

Put in third equation, so these valuessatisfies it.

Hence ^,, where

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