For the system of linear equations x+y+z=6 [ x+ y+7 z=3] x+2 y+3 z=14 which of the followi…

Question

For the system of linear equations

$x+y+z=6$

$$\alpha x+\beta y+7 z=3$$

$x+2 y+3 z=14$

which of the following is NOT true ?

Accepted Answer

$\Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \ \alpha & \beta & 7 \ 1 & 2 & 3 \end{array}\right|$

$$
\begin{aligned}
& =1(3 \beta-14)-1(3 \alpha-7)+1(2 \alpha-\beta) \\
& =3 \beta-14+7-3 \alpha+2 \alpha-\beta \\
& =2 \beta-\alpha-7
\end{aligned}
$$

So, for $\alpha=\beta \neq 7, \Delta \neq 0$ so unique solution.

$\alpha=\beta=7$, equation (i) and (ii) represent 2 parallel planes so no solution.

If $\alpha-2 \beta+7=0$, but $(\alpha, \beta) \neq(7,7)$, then no solution.

Solutions · 1