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?

2. 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?
It isn't so much a theorem as a process.

Consider the graphs of f(x) and g(x). Assume they are both linear functions and the slopes are different, so they cross.

Call the coordinates of the point where they cross (x, y). Then this same point (x, y) satisfies both:
y = f(x)
y = g(x)

So we have that
y = f(x) = g(x)

This is the step that you mentioned.

-Dan