1. Introduction: When the Graph Is Not Enough
You have already met a pair of linear equations. Two equations, two unknowns — usually called x and y. And you have already seen one way to solve them: draw both lines on graph paper and look for the point where they cross.
That works. But here is the part nobody says out loud: the graph only gives you a clean answer when the crossing point lands on neat whole numbers.
Stop scrolling. Try it in your head before reading on. If two lines cross at the point (2.3, 1.7), could you read those numbers off a hand-drawn graph?
You could not. Your pencil is too thick, your eye is not that sharp. You would guess "about 2, about 2" and be wrong.
So we need a method that gives the exact answer every time, using only algebra — no graph paper, no guessing. There are two such methods, and this lesson teaches both:
- Substitution — solve one equation for one variable, then put that into the other.
- Elimination — add or subtract the two equations so that one variable cancels out.
A quick word on what a "solution" even means here. A solution of a pair of equations is a pair of values (one for x, one for y) that makes both equations true at the same time. Not one. Both. By the end of this lesson you will be able to find that pair exactly, choose which method is faster for a given pair, and spot the two strange cases where no single answer exists.