EunoR

2021-02-09

Determine whether the given $\left(2×3\right)\left(2×3\right)$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$x=at+b,y=ct+d,z=et+f$
$2{x}_{1}+{x}_{2}+{x}_{3}=3$
$-2{x}_{1}+{x}_{2}-{x}_{3}=1$

rogreenhoxa8

The given linear equation system
$2{x}_{1}+{x}_{2}+{x}_{3}=3$
$-2{x}_{1}+{x}_{2}+{x}_{3}=1$
The given system of linear equations either represents coincident planes, parallel planes, or coincident planes with a line at their intersection.
Let ${n}_{1}=\left(2,1,1\right)$ and ${n}_{2}=\left(-2,1,-1\right)$ be normal vectors of both the equations. Then n, and na are not parallel.
Let ${x}_{3}=t$, then replace ${x}_{3}=t$ in the system of linear equations

$\left(1\right)+\left(2\right)⇒2{x}_{2}+2t=4⇒2{x}_{2}=4-2t⇒{x}_{2}=2-t$ Replace ${x}_{2}=2-t$ and ${x}_{3}=t$ in the firs equation
$2{x}_{1}+2-t+t=3$
$2{x}_{1}=3-2$
${x}_{1}=\frac{1}{2}$
Hence, the plane intersect in a line and the parametric equations are

Jeffrey Jordon