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Revision guide · IGCSE

IGCSE Maths simultaneous equations: elimination, substitution and the linear-quadratic case

Simultaneous equations are two equations that share the same unknowns and must both be true at once. For two linear equations, use elimination (match the coefficients of one letter, then add or subtract) or substitution. When one equation is linear and the other has a squared term, substitute the linear one into the other, solve the resulting quadratic and find a pair of values for each root. Linear pairs are on every tier of both boards: Cambridge 0580 Core C2.5 and Extended E2.5, Edexcel 4MA1 Foundation and Higher 2.6. The linear and non-linear case is Cambridge Extended (E2.5.5) and Edexcel Higher (2.7D).

Facts checked:

At a glance

Cambridge 0580
Core C2.5.3; Extended E2.5.4, E2.5.5
Edexcel 4MA1
Foundation 2.6A; Higher 2.6, 2.7D
Answer format
Pairs of values, x and y together
Graph link
Solution = intersection point

What each tier expects

Board and tierReferenceWhat can be asked
Cambridge 0580 CoreC2.5.1, C2.5.3Construct and solve two linear equations in two unknowns
Cambridge 0580 ExtendedE2.5.4, E2.5.5Linear pairs, plus one linear and one non-linear equation with powers no higher than two
Edexcel 4MA1 Foundation2.6AExact solution of two linear equations, e.g. 2a + 5b = 12 and 3a + b = 5
Edexcel 4MA1 Higher2.6A, 2.6B, 2.7DLinear pairs, interpreting the solution as an intersection, and linear with quadratic such as x^2 + y^2 = 25

The key ideas

Each linear equation in x and y is a straight line, and the solution of the pair is the point where the two lines cross. That is why there is exactly one pair of values for two different, non-parallel lines, and why a graphical method gives the same answer as the algebra.

Elimination: multiply one or both equations so that one letter has the same coefficient in both. If the signs are the same, subtract the equations; if they are different, add them. The acronym many students use is SSS, Same Signs Subtract. Solve for the letter that remains, then substitute back into either original equation.

Substitution: if one equation already gives y (or x) on its own, such as y = 2x - 11, replace y in the other equation with that expression. This is the only practical method when the second equation contains x^2, y^2 or xy, and it usually produces a quadratic with two roots, so you finish with two pairs of answers. A line can cut a circle or a parabola at two points, touch it at one, or miss it entirely.

Worked example 1: solve 2x + 3y = 17 and 3x - 5y = 35

  1. Make the x coefficients equal: multiply the first equation by 3 to get 6x + 9y = 51, and the second by 2 to get 6x - 10y = 70.
  2. The x terms have the same sign, so subtract: (6x + 9y) - (6x - 10y) = 51 - 70, which gives 19y = -19, so y = -1.
  3. Substitute into the first equation: 2x + 3(-1) = 17, so 2x = 20 and x = 10.
  4. Check in the second equation: 3(10) - 5(-1) = 30 + 5 = 35. Correct.
  5. Answer: x = 10, y = -1.

Worked example 2: solve y = 2x - 11 and x^2 + y^2 = 25

  1. Substitute the line into the circle: x^2 + (2x - 11)^2 = 25.
  2. Expand: x^2 + 4x^2 - 44x + 121 = 25, so 5x^2 - 44x + 96 = 0.
  3. Factorise: (5x - 24)(x - 4) = 0, so x = 4 or x = 4.8.
  4. Find y from the linear equation each time: x = 4 gives y = 8 - 11 = -3, and x = 4.8 gives y = 9.6 - 11 = -1.4.
  5. Check both in the circle: 16 + 9 = 25, and 23.04 + 1.96 = 25. Answer: (4, -3) and (4.8, -1.4).

Worked example 3: forming the equations from words

  1. Three pens and two rulers cost $4.30. One pen and two rulers cost $2.10. Let a pen cost p dollars and a ruler r dollars.
  2. Write the equations: 3p + 2r = 4.30 and p + 2r = 2.10.
  3. The r terms match, so subtract: 2p = 2.20, giving p = 1.10.
  4. Substitute: 1.10 + 2r = 2.10, so r = 0.50.
  5. Check: 3(1.10) + 2(0.50) = 3.30 + 1.00 = 4.30. A pen costs $1.10 and a ruler $0.50.

Common mistakes that cost marks

  • Multiplying only one side of an equation, or only some of its terms, when matching coefficients.
  • Subtracting when the signs differ: with -10y and +9y you add if you want the y terms to cancel.
  • Arithmetic with negatives in the subtraction step: 51 - 70 is -19, not 19.
  • In the linear-quadratic case, giving the x values only, or pairing an x with the wrong y. Always use the linear equation to find y.
  • Squaring a bracket wrongly: (2x - 11)^2 is 4x^2 - 44x + 121, not 4x^2 + 121.
  • Not checking: substituting into the equation you did not use catches nearly every slip.

Exam technique and how a tutor helps

Linear simultaneous equations are reliable marks, usually 3 or 4, and appear on non-calculator papers, so practise them without a calculator. Show the scaled equations and the add or subtract line: if you then slip in the arithmetic, those lines still earn method marks. The linear-quadratic version is a longer question, typically 5 marks, and examiners expect two complete coordinate pairs.

When a question uses words, define your letters in a sentence first. It forces you to decide what each letter means and avoids mixing pens with rulers halfway through.

A tutor working one to one can see which of the three common breakdowns a student has: choosing the wrong operation, losing a negative, or getting lost in the quadratic expansion. Lessons target that breakdown with short sets of graded questions, then move to past-paper problems where the student has to build the equations from a context such as ticket prices or ages.

Self-check: can you do these?

  • Solve x + y = 14 and x - y = 2. (Answer: x = 8, y = 6)
  • Solve 2a + 5b = 12 and 3a + b = 5. (Answer: a = 1, b = 2)
  • Solve y = x^2 and y = x + 6. (Answer: (3, 9) and (-2, 4))
  • Solve y = 11x - 2 and y = 5x^2. (Answer: (2, 20) and (0.2, 0.2))
  • Explain what the solution of two linear simultaneous equations looks like on a graph.

Common questions

Should I use elimination or substitution?

For two linear equations in the form ax + by = c, elimination is usually quicker and less error-prone. If one equation already gives y = ... or x = ..., or if the other equation has a squared term, use substitution.

Are linear-quadratic simultaneous equations on the Core or Foundation paper?

No. They are Cambridge 0580 Extended (E2.5.5) and Edexcel 4MA1 Higher (2.7D). Linear pairs are on every tier of both boards.

Why do I get two answers for x in the harder questions?

Because substituting a line into a curve gives a quadratic, and a line can meet a curve at two points. Each x value gives its own y value, so you finish with two coordinate pairs.

Can I solve simultaneous equations by drawing graphs?

Yes, and some questions ask for that. The answer is the intersection point, read to the accuracy of the grid. An algebraic method gives exact values, so use it unless the question says to use the graph.

What does a LiveTutor maths lesson cost?

Lessons are one to one, online and 60 minutes, at $15 a lesson for any subject or level. You choose 1 to 5 lessons a week on a monthly plan, and the first lesson is a free trial.

Sources

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