Foundations — Simultaneous equations — substitution, elimination, cross-multiplication

Before you can play with substitution, elimination, or cross-multiplication, you need to own every piece of notation the parent note quietly assumes. Below we build each one from nothing — plain words first, then a picture, then the reason the topic can't live without it.

1. The unknown letters $x$ and $y$

The picture: think of $x$ as the horizontal position and $y$ as the vertical position of a dot on a flat sheet of graph paper. A pair $(x,y)$ is a single dot's address.

Why the topic needs it: simultaneous equations have two unknowns. We need two separate names so we can talk about them independently — and so we can later "kill off" one of them and solve for the other.

2. Coefficients $a$, $b$ and constants $c$ (and the subscripts)

Look at the parent's general system: $a_{1}x+b_{1}y=c_1{a}_{2}x+b_{2}y=c_2$

Four kinds of symbol are hiding here. Let's name each.

The picture: imagine two shopping receipts. Receipt 1 says "$a_1$ apples at fixed weight plus $b_1$ bananas = total $c_1$". Receipt 2 is a different shop with its own prices $a_2,b_2,c_2$. The subscript is just the receipt number.

Why the topic needs it: with two equations we have two of everything. Subscripts let us write the general recipe once instead of inventing brand-new letters for every problem.

3. What "$=$" really promises, and what a solution is

The picture: a see-saw perfectly level. If you add 14 to the left plate, you must add 14 to the right plate or it tips. Every legal algebra move is "same thing to both plates".

Why the topic needs it: every step in substitution and elimination — "add 14 to both sides", "multiply the whole equation by $a_2$" — is only allowed because $=$ is a balance. This is the licence for all the moves.

4. A linear equation = a straight line

The picture: pick any $x$, the rule forces one matching $y$. Sweep $x$ across the sheet and the matching dots trace a ruler-straight path.

Why the topic needs it: the whole "two lines" intuition of the parent note only works because each equation is a line. If you fully understand this, see Linear Equations in Two Variables for a single line's behaviour, and Graphical Method for Simultaneous Equations for finding the crossing by eye.

5. Two lines together — the three possible pictures

Two straight lines on one sheet can relate in exactly three ways. The parent note lists them; here is what each looks like.

Why the topic needs it: every method later hands you a fraction. When that fraction's bottom is zero, you're secretly in the parallel or identical case — the algebra is telling you the lines never cross uniquely.

6. The denominator $a_1{b}_2-a_2{b}_1$ — meet the determinant

The parent note keeps bumping into the same bottom of a fraction: $a_1{b}_2-a_2{b}_1$. Let's demystify it.

Why the topic needs it: this one number is the gatekeeper of every method. When you're ready to go deeper, this idea grows into Determinants, powers Cramer's Rule, and generalises through Matrix Methods.

7. Standard form (and why methods "prefer" it)

Why the topic needs it: cross-multiplication reads coefficients by position. If your equation is scrambled (e.g. $y=7-2x$), you must first tidy it to $2x+y=7$ so the letters line up under $a$, $b$, $c$. Substitution, by contrast, is happy with the messy isolated form.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, re-read that section above.