Sometimes we want to calculate the line at which two planes intersect each other. We can accomplish this with a system of equations to determine where these two planes intersect. Note that this will result in a system with parameters from which we can determine parametric equations from.

# Lines of Intersection Between Planes

Let's hypothetically say that we want to find the equation of the line of intersection between the following lines $L_1$ and $L_2$:

(1)We will begin by first setting up a system of linear equations. Note that we have more variables (3) than the number of equations (2), so there will be a column of zeroes after we convert the matrix of lines $L_1$ and $L_2$ into reduced row echelon form. The following matrix represents our two lines: $\begin{bmatrix}2 & -1 & -4 & -2 \\ -3& 2 & -1 & -2 \end{bmatrix}$.

We will thus convert this matrix intro reduced row echelon form by Gauss-Jordan Elimination:

(2)We now have the system in reduced row echelon form. We can see that we have a free parameter for $z$, so let's parameterize this variable. Let $z = t$ for $(-\infty < t < \infty)$. Therefore, we can determine the equation of the line as a set of parameterized equations:

(7)