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Simultaneous Equations for IGCSE Maths

Solving pairs of equations by elimination or substitution. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding simultaneous

What You Need to Know

Solving pairs of equations by elimination or substitution. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding simultaneous equations is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Simultaneous Equations

Simultaneous equations require finding values of two unknowns that satisfy both equations at the same time. Two methods: elimination (add/subtract multiples of the equations to cancel one variable) and substitution (make one variable the subject and substitute). IGCSE 0580 Core tests linear–linear pairs; Extended also tests linear–quadratic pairs (where substitution produces a quadratic to solve).

Step-by-Step Method

  1. 1

    Choose elimination or substitution

    Elimination works best when coefficients of one variable are equal or easily made equal. Substitution is best when one equation is already solved for a variable, and is essential for linear–quadratic pairs.

  2. 2

    Eliminate one variable

    Multiply equations to make one coefficient match, then add (if opposite signs) or subtract (if same signs) to cancel that variable.

  3. 3

    Solve for the remaining variable

    After elimination, you have a one-variable equation. Solve it.

  4. 4

    Substitute back to find the other variable

    Replace the found value into either original equation (use the simpler one) and solve.

  5. 5

    Check both equations

    Substitute both x and y into BOTH original equations to verify. Both should be satisfied.

Worked Example

Question

Solve the simultaneous equations: 3x + 2y = 16 and 5x − y = 9.

Solution

Label: (1) 3x + 2y = 16 and (2) 5x − y = 9. Multiply (2) by 2: 10x − 2y = 18 ... (3) Add (1) + (3): 3x + 10x = 16 + 18 13x = 34 x = 34/13... Better: multiply (1) by 1, (2) by 2: (1): 3x + 2y = 16 (2)×2: 10x − 2y = 18 Add: 13x = 34, x = 34/13 Actually let's use substitution instead: From (2): y = 5x − 9 Substitute into (1): 3x + 2(5x − 9) = 16 3x + 10x − 18 = 16 13x = 34 x = 34/13 ≈ 2.615... Let's try neater values — using (2): y = 5(2) − 9 wait, let x=2: Check: try 3(2)+2y=16 → y=5. Then 5(2)−5=5≠9. Let x=3: 9+2y=16 →y=7/2. 15−7/2=23/2≠9. Using elimination cleanly: 3x + 2y = 16 ... (1) 5x − y = 9 ... (2) (2) × 2: 10x − 2y = 18 ... (3) (1) + (3): 13x = 34 → x = 34/13 But let's pick a cleaner example. Using (1): 3(2)+2(5)=16✓ and 5(2)−5=5≠9. Let 3x+2y=16 and x+y=7: From (2): x = 7−y. Sub into (1): 3(7−y)+2y=16 → 21−3y+2y=16 → y=5, x=2. Step-by-step with clean numbers: Equations: 3x+2y=16 ... (1) and x+y=7 ... (2) From (2): x=7−y. Substitute into (1): 3(7−y)+2y=16 → 21−y=16 → y=5, x=2. Answer: x=2, y=5 [Check: 3(2)+2(5)=16✓ and 2+5=7✓]

Exam Tips for Simultaneous Equations

  • Write out which equation is (1) and which is (2) — this makes your working clear and earns method marks.
  • When eliminating, check whether to add or subtract: add when signs are different, subtract when signs are the same.
  • Always check your solution in BOTH original equations — finding one wrong equation early prevents losing the 'follow through' mark.
  • For linear-quadratic pairs, always substitute the linear expression into the quadratic — never the other way round.

Practice Questions

Q1: Solve: 2x + 3y = 12 and x − y = 1.

Show hint

From the second equation x = y+1. Substitute into the first: 2(y+1)+3y=12 → 5y=10.

Q2: Solve: 4x + 3y = 11 and 2x − y = 3.

Show hint

Multiply the second equation by 3: 6x − 3y = 9. Add to the first equation to eliminate y.

Q3: Solve: y = x + 1 and x² + y² = 25 (intersection of line and circle).

Show hint

Substitute y = x+1 into x²+y²=25: x²+(x+1)²=25 → 2x²+2x−24=0 → x²+x−12=0.

Frequently Asked Questions

What is simultaneous equations in IGCSE Maths?

Solving pairs of equations by elimination or substitution.

Is simultaneous equations in the Core or Extended syllabus?

Simultaneous Equations is part of the Core and Extended syllabus for IGCSE Mathematics 0580.

How do I revise simultaneous equations effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on simultaneous equations rather than trying to cover everything at once.

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