Question: Consider the three planes
$P_{1}: 3 x+15 y+21 z=9$
$P_{2}: x-3 y-z=5$, and
$P_{3}: 2 x+10 y+14 z=5$
Then, which one of the following is true?
(1) $\mathrm{P}_{1}$ and $\mathrm{P}_{3}$ are parallel.
(2) $\mathrm{P}_{2}$ and $\mathrm{P}_{3}$ are parallel.
(3) $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ are parallel.
(4) $\mathrm{P}_{1}, \mathrm{P}_{2}$ and $\mathrm{P}_{3}$ all are parallel.
Correct Option: 1
Solution:
$P_{1}=x+5 y+7 z=3$
$P_{2}=x-3 y-z=5$
$P_{3}=x+5 y+7 z=5 / 2$
$\Rightarrow P_{1} \| P_{3}$
