# How do matrix solutions represent an intersection of lines?

I'm very new to linear algebra, and I have a homework problem that hasn't been covered in the book or by the professor. It seems like I have a fundamental misunderstanding of what matrices represent, but I can't find a good article or answer.

Do the three lines $x_1 - 4x_2 = 1$, $2x_1 - x_2 = -3$, and $-x_1 - 3x_2 = 4$ have a common point of intersection? Explain.

I assumed that the solution set of the matrix would represent how many intersections there were. I solved the echelon form and got: $$\begin{bmatrix}1 & -4 & & 1\\2& -1 & & -3\\-1 & -3 & & 4\end{bmatrix} \rightarrow \begin{bmatrix}1 & -4 & & 1\\0& 1 & & -\frac{5}{7}\\0 & 0 & & 0\end{bmatrix}$$

Since this has infinite solutions, I would have thought it meant there were infinite intersections, or rather two equivalent lines, but that obviously isn't true. Is there any relationship between the solution set of a matrix and its original equations/lines? What is the matrix actually representing?

You can reduce it further. When you do, the first row is $(1,0,-13/7)$ which says that your simultaneous solution is $(-13/7, -5/7)$. So, there is one point of intersection.
Your solution is correct (I assume you have row-reduced properly) but the bottom row gives you no information - does $0x + 0y + 0z = 0$ tell you anything about $x, y,$ or $z$? You can safely scrub out a row of zeroes from a matrix and solve the system from there.
If your augmented matrix is of the form $$[A | C]$$ then system has unique solution if $$rank(A)=rank(C)$$=number of unknowns. Here, $$ank(A)=rank(C)=$$number of unknowns=2 So, unique soon i.e a single intersection point