1. Introduction: When the Equation Looks Scary but Isn't
You have spent this whole chapter solving pairs of linear equations. Things like 2x + 3y = 13. Clean. Friendly. The unknowns sit by themselves, no fractions, no x stuck under a line.
Now look at this pair:
2/x + 3/y = 13 5/x − 4/y = −2
Your first reaction might be, "This is not linear. x and y are in the denominator. I have not been taught this." And you are right that it is not linear — not yet. But here is the idea this whole lesson rests on: some equations are not linear, but they are wearing a disguise. Underneath, they are linear. We just have to spot the disguise and take it off.
The trick is a small, almost silly move. What if we gave the messy part a new, simple name?
Stop scrolling. Try it in your head before reading on. In the pair above, what single thing repeats in both equations?
It is 1/x and 1/y. Both equations are built out of those two pieces. So let us call 1/x by the name "u", and 1/y by the name "v". The moment we do that, the scary pair becomes:
2u + 3v = 13 5u − 4v = −2
That is just a normal linear pair in u and v. You already know four ways to crack it. Solve for u and v, then walk backwards to x and y.
That is the entire topic. We will go slow, because the slips here are small and sneaky, but the big idea is exactly this: rename the repeating piece, solve the easy linear pair, then go back.
By the end of this lesson you will be able to spot when an equation can be made linear, choose the right substitution, solve the new linear pair, and — the step everyone forgets — convert back to the original unknowns.