1. Introduction: One Formula That Solves Any Pair
By now you have three ways to handle a pair of linear equations. You can draw the two lines and look for where they cross. You can use substitution. You can use elimination. They all work — but each one needs you to think a little, and on a bad day the thinking is where mistakes creep in.
Now imagine a different deal. What if there were a single formula — write the numbers into a fixed pattern, do a fixed set of multiplications, and out comes x and y every single time? No deciding which variable to eliminate. No guessing what to substitute. Just slot the numbers in.
That is the cross-multiplication method. It is a ready-made formula for solving any pair of linear equations in two variables.
Here is the one thing it asks of you. Both equations must first be written in the same tidy shape:
ax + by + c = 0
Everything moved to the left, equals zero on the right. That shape is called the standard form. Once both equations are in standard form, the formula takes over.
Stop scrolling. Try it in your head before reading on. The equation 3x + 2y = 7 — can you move it into the shape "something = 0"?
3x + 2y − 7 = 0. You just shifted the 7 across. That is all "standard form" means: bring the constant to the left so the right side is 0.
By the end of this lesson you will be able to write a pair of equations in standard form, read off their six numbers, drop them into the cross-multiplication formula, and get x and y — and you will know the one situation where the formula refuses to give an answer.