The following system of linear equations 7x + 6y - 2z = 0 3x + 4y +...

The following system of linear equations

7x + 6y - 2z = 0

3x + 4y + 2z = 0

x - 2y - 6z = 0,

has

Options

Ainfinitely many solutions, $\left(x, y, z\right)$ satisfying $y = 2z$
Bno solution
Cinfinitely many solutions, $\left(x, y, z\right)$ satisfying $x = 2z$
Donly the trivial solution

Correct answer

C. infinitely many solutions, $\left(x, y, z\right)$ satisfying $x = 2z$

Step-by-step solution

$\left|$ \begin{cases}7 & 6 & - 2 \\ 3 & 4 & 2 \\ 1 & - 2 & - 6\end{cases} $\right|$

$= 7 \left(- 20\right) - 6 \left(- 20\right) - 2 \left(- 10\right)$ = - 140 + 120 + 20 = 0 $So infinite non-trivial solution exist Now equation$ ( 1 ) + 3 $equation$ ( 3 ) $ $ ⇒ 10x - 20z = 0 $ ⇒ x = 2z$

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