Originally Posted by

**earachefl** Here's something I found interesting. "If $\displaystyle f(x) = 3x + 7$ and $\displaystyle g(x) = -2x + 5$, find the abscissa of a point which is on the graphs of both $\displaystyle f$ and $\displaystyle g$."

I really didn't know how to do it; the book certainly didn't spell it out anywhere. I started by graphing both equations, getting a rough idea of where the equations crossed, and plugged in a couple of points until I found the point $\displaystyle (-\frac{2}{5}, 5\frac{4}{5})$. Then I just happened to notice that earlier I had tried simplifying $\displaystyle 3x + 7 = -2x + 5$, which simplifies to $\displaystyle 5x = -2$, and that those numbers corresponded to the abscissa of the point in question. (Never mind that they also correspond to the numbers of the function $\displaystyle g(x)$).

I tried a few other simple linear equations and got the same results, so I feel like I've stumbled upon a consistent solution to the problem. Obviously I haven't discovered anything new, but I wonder, is this solution valid to all sets of two linear equations? If so, is there a named theorem for it?