Simultaneous Equations

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Edexcel IAL · Pure 1

Simultaneous Equations

Two equations, two unknowns — find the values that make both true at the same time. Master elimination, substitution, and the quadratic case.

3 Methods Elimination · Substitution · Quadratic Works Offline

The Big Idea

A single equation with two unknowns has infinitely many solutions. But give me two equations and those two lines cross at exactly one point — and that crossing point is the answer. Simultaneous equations is just finding where two lines meet.

Analogy

Imagine two friends walking along two different straight roads on a flat map. The question "where do they meet?" is exactly a simultaneous equation. If the roads are parallel they never meet (no solution). If they're the same road, they always meet (infinite solutions). Otherwise — one crossing point.

Chapter Overview

Elimination

Scale one or both equations so a variable cancels when you add or subtract. Best when coefficients are easy to match.

Substitution

Rearrange one equation to isolate a variable, then substitute into the other. Required for quadratic pairs.

Quadratic Pairs

One linear + one quadratic equation. Use substitution to get a single quadratic, then solve for usually two solution pairs.

Method 1: Elimination

Make one variable disappear by adding or subtracting the equations

How It Works

The core idea is beautifully simple: if you can get the same number in front of one variable in both equations, you can cancel it out completely. You're left with one equation, one unknown — easy.

Think of it like this: if equation ① has 4y and equation ② also has 4y, then ① − ② makes the y's vanish, leaving only x to solve.

The Elimination Rule

Same signs → Subtract | Different signs → Add

After eliminating, solve for one variable, then substitute back.

Step-by-Step Process

Label your equations ① and ②. Keeps track of what you're doing and earns marks.
Scale one (or both) equations by multiplying by a constant so the coefficients of one variable match in both equations.
Add or subtract the equations. Same signs → subtract. Different signs → add. This eliminates one variable.
Solve the resulting single-variable equation to find the value of that unknown.
Substitute back into one of the original equations to find the other variable.
Check: substitute both values into the equation you haven't used yet. If it balances, you're done.

Examiner's Warning: Mishandling minus signs is the #1 cause of errors in this method. Be extra careful when subtracting — subtract every term, not just the one you're eliminating.

Worked Example

Worked Example — Elimination

Solve the simultaneous equations:
−2x + 4y = 5 ①
4x − 5y = −7 ②

STEP 1

Multiply equation ① by 2 to match the x-coefficients:
2 × ①: −4x + 8y = 10 ② stays: 4x − 5y = −7 The x-coefficients are now −4 and +4 — different signs, so we add.

STEP 2

Add the two equations together (different signs → add): (−4x + 8y) + (4x − 5y) = 10 + (−7) 3y = 3

STEP 3

STEP 4

STEP 5

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Also in the full note

Method 2: Substitution
Method 3: Quadratic Simultaneous Equations

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Simultaneous Equations

Simultaneous Equations

Method 1: Elimination

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