Question

# Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two

Forms of linear equations

Determine whether the given $$(2×3)(2×3)$$ 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
$$x=at+b , y=ct+d , z=et+f$$
$$2x_1+x_2+x_3=3$$
$$-2x_1+x_2-x_3=1$$

2021-02-10
The given system of linear equations
$$2x_1+x_2+x_3=3$$
$$-2x_1+x_2+x_3=1$$
We need to determine whether the given system of linear equations represents coincident planes, two parallel planes, or two planes whose intersection is a line.
Let $$n_1 = (2,1,1)$$ and $$n_2 = (-2,1,-1)$$ 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 $$2x_1+x_2+t=3 \ \ \ (1)$$
$$-2x_1+x_2+t=1 \ \ \ (2)$$
$$(1)+(2) \Rightarrow 2x_2+2t=4 \Rightarrow 2x_2=4-2t \Rightarrow x_2=2-t$$ Replace $$x_2 = 2-t$$ and $$x_3 =t$$ in the firs equation
$$2x_1+2-t+t=3$$
$$2x_1=3-2$$
$$x_1=\frac{1}{2}$$
Hence, the plane intersect in a line and the parametric equations are
$$x_1=\frac{1}{2} , x_2=2-t \text{ and } x_3=t$$