Simultaneous Equations through Elimination

Teacher’s Guide: Delivering the Simultaneous Equations Lesson

This section provides a walkthrough of the full lesson, including editable PowerPoint slides, differentiated worksheets, and teaching strategies refined over 20 years of classroom practice.

Below, you will find sample materials from the lesson alongside a step-by-step teaching guide covering how to introduce elimination, structure independent practice, and address common sign errors.

Once students have secured elimination, members can extend learning with Simultaneous Equations by Substitution and a dedicated problem-solving lesson that applies simultaneous equations in unfamiliar contexts.

Differentiated Learning Objectives

• 🟡 All students should solve a pair of equations simultaneously using elimination when the coefficients of one unknown are equal.
• 🟢 Most students should solve a pair of equations simultaneously using elimination when one coefficient is a factor of the other.
• 🟣 Some students should derive and solve a pair of equations simultaneously using elimination where one coefficient is a factor of the other.

Step-by-Step Method for Simultaneous Equations

Use this approach for linear simultaneous equations in GCSE maths (elimination method):

1. Label the equations A and B.
2. Multiply one or both equations so one variable has matching coefficients.
3. Add or subtract to eliminate that variable, then solve and substitute back to check.

Quick reference: elimination vs substitution

⚠️ Common mistakes

• ❌ Not multiplying every term — scale the right-hand side constant as well as the x and y terms.
• ❌ Adding when you should subtract — same signs → subtract; opposite signs → add.
• ❌ Sign errors with double negatives — subtracting a negative produces a positive.
• ❌ Skipping the check — substitute both values into a different original equation from the one used to find the second unknown.

Teaching Simultaneous Equations

Teaching students how to solve simultaneous equations works well across three lessons. In Lesson 1, focus on traditional questions with coefficients of the same magnitude. This helps students understand why we eliminate variables — and why y + (−y) simplifies to zero by addition, while y − (−y) gives 2y, not zero.

In Lesson 2, tackle questions that require scaling one or both equations to match coefficients before eliminating. In Lesson 3, students construct simultaneous equations from geometric diagrams and real-life contexts. Starting with the diagram before the algebra prevents students from losing sight of what the variables represent.

• Elimination: match coefficients, then add or subtract (most common on GCSE papers).
• Substitution: replace a variable when one equation already gives y = … or x = …
• Teacher tip: give students a printed handout of the example you are modelling so they annotate rather than copy from the board.

By the end of a worked example, students should have a correctly annotated solution to reference during independent practice — including which equations were labelled A and B, which coefficient was matched, and whether the equations were added or subtracted.

Worked example: elimination

Solve 5x + 2y = 24 and 2x − y = 6 (matching Example A from the video).

Multiply equation B by 2 so the y-coefficients match in magnitude with opposite signs:

Add to eliminate y: 9x = 36 → x = 4. Substitute into A: 5(4) + 2y = 24 → y = 2. Check in B: 2(4) − 2 = 6 ✓.

When to add and when to subtract

• Add when the matching coefficients have opposite signs — for example +2y and −2y.
• Subtract when the matching coefficients have the same sign — for example +2x in both equations.

A useful check: after eliminating, only one variable should remain. If both disappear, the equations are multiples of each other. If neither disappears, the coefficients have not been matched correctly.

Solving by substitution

When one equation already gives y (or x) as the subject — such as y = 14 − 3x — substitution is faster than elimination. Replace y in the other equation with the expression, then solve.

For example, if equation A is 2x − 5y = 32 and equation B is y = 14 − 3x, substitute B into A: 2x − 5(14 − 3x) = 32 → 17x = 102 → x = 6. Then y = 14 − 3(6) = −4.

💡 Teacher tips for this lesson

• Starter: put two equations on the board and ask students to decide whether they would add or subtract — before calculating.
• Support: begin with equal coefficients of the same magnitude so students secure the sign rules first.
• Challenge: set word problems and geometry diagrams where students must define variables before forming equations.
• Common fix: if students forget to multiply the constant, ask them to rewrite the scaled equation with every term circled.