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For the System of Equations: X + 2y + 3z = 1 2x + Y + 3z = 2 5x + 5y + 9z = 4 (A) There is Only One Solution (B) There Exists Infinitely Many Solution (C) There is No Solution (D) None of These

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Question

For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4

Options

  • there is only one solution

  • there exists infinitely many solution

  • there is no solution

  • none of these

MCQ
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Solution

(a) there is only one solution
The given system of equations can be written in matrix form as follows:
\[\begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 2 \\ 4\end{bmatrix}\]
Here,
\[A=\begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{bmatrix},X=\begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}1 \\ 2 \\ 4\end{bmatrix}\]
Now, 
\[\left| A \right| = \begin{vmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{vmatrix}\]
\[ = 1\left( 9 - 15 \right) - 2\left( 18 - 15 \right) + 3\left( 10 - 5 \right)\]
\[ = - 6 - 6 + 15\]
\[ = 3 \neq 0\]
\[ \Rightarrow \left| A \right|\neq 0 \]
So, the given system of equations has a unique solution.

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Chapter 7: Solution of Simultaneous Linear Equations - Exercise 8.4 [Page 22]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 7 Solution of Simultaneous Linear Equations
Exercise 8.4 | Q 8 | Page 22

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