This is a *System of Linear Equations (SLE)*. Solving it, we get \(s=0\) and \(t=1\). This means that the intersection of \(\ell_1\) and \(\ell_2\) is the one point \(P\) with \[
\vec{OP}=\mathbf r_1(s=0)=\mathbf r_2(t=1)=\left(\begin{smallmatrix}2\\-1\\0\end{smallmatrix}\right).
\]

*Example*. Let's now try to find the intersections of the lines \[
\ell_1:\mathbf r_1(t)=\left(\begin{smallmatrix}0\\2\\3\end{smallmatrix}\right)+t\left(\begin{smallmatrix}1\\-1\\3\end{smallmatrix}\right)\text{ and }
\ell_2:\mathbf r_2(t)=\left(\begin{smallmatrix}1\\0\\1\end{smallmatrix}\right)+t\left(\begin{smallmatrix}1\\1\\1\end{smallmatrix}\right).
\] In this case, there are no solutions. Also, the direction vectors are not parallel: there is no scalar \(c\) such that \(c\left(\begin{smallmatrix}1\\-1\\3\end{smallmatrix}\right)=\left(\begin{smallmatrix}1\\1\\1\end{smallmatrix}\right)\). Therefore, in this case, \(\ell_1\) and \(\ell_2\) are *skew lines*. This is a situation new to \(m\)-space with \(m>2\): in the plane, every pair of nonparallel lines intersect. Actually, this is the *generic* situation: if you randomly select two lines, they will almost always be skew.

*Example*. Let's now try to find the intersections of the lines \[
\ell_1:\mathbf r_1(t)=\left(\begin{smallmatrix}1\\2\\0\end{smallmatrix}\right)+t\left(\begin{smallmatrix}-1\\2\\1\end{smallmatrix}\right)\text{ and }
\ell_2:\mathbf r_2(t)=\left(\begin{smallmatrix}0\\1\\-2\end{smallmatrix}\right)+t\left(\begin{smallmatrix}2\\-4\\-2\end{smallmatrix}\right).
\] There are no solutions. Also, the two lines are parallel: we have \(-2\left(\begin{smallmatrix}-1\\2\\1\end{smallmatrix}\right)=\left(\begin{smallmatrix}2\\-4\\-2\end{smallmatrix}\right)\).

*Example*. Let's now try to find the intersections of the lines \[
\ell_1:\mathbf r_1(t)=\left(\begin{smallmatrix}-3\\0\\2\end{smallmatrix}\right)+t\left(\begin{smallmatrix}-1\\2\\1\end{smallmatrix}\right)\text{ and }
\ell_2:\mathbf r_2(t)=\left(\begin{smallmatrix}0\\1\\-2\end{smallmatrix}\right)+t\left(\begin{smallmatrix}2\\-4\\-2\end{smallmatrix}\right).
\] Here, we can see that the two lines are actually the same: the two direction vectors are parallel, and also the vector between the two starting points is parallel to the direction vectors. Therefore, the solution set is a *1-parameter collection*, which in this case can be either of the parametrizations we're already given: \[
P(t)=(-1+2t,-4-4t,-2t).
\]

is not a SLE, because the third equation has the nonlinear summand \(x_2x_3\).

A SLE is inTo solve a SLE, we want to perform operations on it which don't change the set of solutions. We can use the following *Elementary Operations*: 1. Swap two rows 2. Multiply an equation by a nonzero scalar 3. Add to an equation a scalar multiple of another equation

We have replaced \(z\) by \(t\) on the right sides for the sake of clarity.

*Example*. Let's find the intersections of the planes with equations \[
\Pi_1:-3x+y-2z=2\text{ and }\Pi_2:6x-2y+4z=3.
\] Here, there are no solutions. Note that the normal vectors of the two planes are parallel.

The two planes are actually the same.