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

\(x_1=\frac{1}{2} , x_2=2-t \text{ and } x_3=t\)

\(2x_1+x_2+x_3=3\)

\(-2x_1+x_2+x_3=1\)

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

\(x_1=\frac{1}{2} , x_2=2-t \text{ and } x_3=t\)