Proportion, simultaneous equations, quadratics and inequalities: these are the topics that separate a comfortable pass from a strong grade. Here is how they work.
If you have made it through the first half of the Equations, formulae and identities section, you already know how to manipulate expressions, substitute into formulae, and solve linear equations. Good. Now the Pearson Edexcel IGCSE Mathematics Specification A syllabus raises the bar. This second half introduces proportion (Higher only), simultaneous linear equations, quadratic equations, and inequalities. These topics show up consistently on both papers of the 4MA1 exam, and the marks they carry are significant.
These edexcel igcse mathematics specification a revision notes walk through each topic with worked examples, the kind of detail you actually need for the exam, and honest warnings about where students drop marks. If you are looking for edexcel igcse mathematics specification a notes on the trickier parts of algebra, you are in the right place.
Proportion (Higher tier)
Direct and inverse proportion are Higher tier topics, but they are examined regularly. The specification expects you to set up algebraic relationships and link them to graphs.
Direct proportion
If y is directly proportional to x, we write y is proportional to x, which means y = kx. If y is directly proportional to x2, then y = kx2. You use one pair of values to find k, then apply the formula to find other values.
y = kx2
36 = k(3)2 = 9k
k = 36/9 = 4
So y = 4x2
When x = 5: y = 4(25) = 100
Inverse proportion
If y is inversely proportional to x, then y = k/x. As x increases, y decreases. If y is inversely proportional to x2, then y = k/x2.
y = k / sqrt(x)
8 = k / sqrt(9) = k / 3
k = 24
When x = 25: y = 24 / sqrt(25) = 24 / 5 = 4.8
The edexcel exam often asks you to interpret the graph of a proportion relationship. Direct proportion to x gives a straight line through the origin. Direct proportion to x2 gives a parabola through the origin. Inverse proportion to x gives a reciprocal curve. Knowing the shape helps you identify the type of relationship before you even start calculating.
Simultaneous linear equations
A pair of simultaneous equations has two unknowns, and you need both values. The igcse specification covers solving these algebraically. There are two methods: elimination and substitution. Both give the same answer, so use whichever feels more natural to you.
Elimination method
The y-coefficients are +2 and -2. Adding the equations eliminates y:
(3x + 2y) + (5x - 2y) = 16 + 24
8x = 40
x = 5
Substitute x = 5 into the first equation:
3(5) + 2y = 16
15 + 2y = 16
2y = 1
y = 0.5
Solution: x = 5, y = 0.5
Check in second equation: 5(5) - 2(0.5) = 25 - 1 = 24. Correct.
Substitution method
Substitute y = 2x - 1 into the second equation:
3x + (2x - 1) = 14
5x - 1 = 14
5x = 15
x = 3
y = 2(3) - 1 = 5
Solution: x = 3, y = 5
At Higher tier, you also need to recognise that solving simultaneous equations graphically means finding the point where two lines intersect. If you plot both equations and they cross at (3, 5), that is your solution.
Quadratic equations
A quadratic equation has the form ax2 + bx + c = 0. The edexcel specification requires solving by factorisation at both tiers (Foundation is limited to cases where a = 1), and at Higher tier, by the quadratic formula and by completing the square.
Solving by factorisation
Factorise: find two numbers that multiply to 6 and add to -5.
-2 x -3 = 6, and -2 + (-3) = -5. Yes.
(x - 2)(x - 3) = 0
Either x - 2 = 0, so x = 2
Or x - 3 = 0, so x = 3
Solutions: x = 2 or x = 3
The quadratic formula (Higher)
When factorisation is not straightforward, use the formula: x = (-b +/- sqrt(b2 - 4ac)) / 2a. This formula is given on the formulae sheet, so you do not need to memorise it, but you do need to use it accurately.
a = 2, b = 3, c = -7
Discriminant = b2 - 4ac = 9 - 4(2)(-7) = 9 + 56 = 65
x = (-3 + sqrt(65)) / (2 x 2) or x = (-3 - sqrt(65)) / (2 x 2)
x = (-3 + 8.0623...) / 4 or x = (-3 - 8.0623...) / 4
x = 5.0623.../4 or x = -11.0623.../4
x = 1.27 or x = -2.77 (to 2 d.p.)
A common error is forgetting the +/- in the formula, giving only one solution. Quadratic equations almost always have two solutions. If the discriminant (b2 - 4ac) is positive, there are two distinct real roots. If it is zero, there is one repeated root. If it is negative, there are no real roots.
Simultaneous equations: one linear, one quadratic (Higher)
At Higher tier, you may need to solve a linear equation and a quadratic equation simultaneously. The method is always substitution: rearrange the linear equation to express one variable, then substitute it into the quadratic.
Substitute y = x + 1 into x2 + y2 = 13:
x2 + (x + 1)2 = 13
x2 + x2 + 2x + 1 = 13
2x2 + 2x + 1 = 13
2x2 + 2x - 12 = 0
x2 + x - 6 = 0
(x + 3)(x - 2) = 0
x = -3 or x = 2
When x = -3: y = -3 + 1 = -2
When x = 2: y = 2 + 1 = 3
Solutions: (-3, -2) and (2, 3)
Inequalities
Inequalities use the symbols < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to). You solve them almost exactly like equations, with one critical difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
Solving linear inequalities
3x - 7 > 5
3x > 12
x > 4
On a number line, this is an open circle at 4 with an arrow pointing right.
-2x <= 8
x >= -4 (inequality reversed because we divided by -2)
On a number line, this is a filled circle at -4 with an arrow pointing right.
Representing inequalities on a number line
Open circles mean "not including" (strict inequality: < or >). Filled circles mean "including" (<= or >=). The convention is important: the edexcel igcse mark scheme specifically penalises using the wrong circle type.
Graphical inequalities
At Foundation and Higher tier, you need to shade regions on a coordinate grid that satisfy given inequalities. For example, y < 2x + 1 means the region below the line y = 2x + 1. Draw the boundary line (dashed for strict inequality, solid for <= or >=), then shade the correct side.
Quadratic inequalities (Higher)
To solve x2 - 5x + 6 < 0, first factorise to get (x - 2)(x - 3) < 0. The critical values are x = 2 and x = 3. Sketch the quadratic (a U-shape crossing the x-axis at 2 and 3). The curve is below zero between the roots, so the solution is 2 < x < 3.
Common mistakes
- Not reversing the inequality when dividing by a negative. This is the single most common error in inequality questions. -3x > 12 becomes x < -4, not x > -4.
- Giving only one solution to a quadratic. If the question says "solve," you need both roots. If it says "find the positive solution," then one root is fine. Read the question.
- Rounding before the final step in the quadratic formula. Calculate the full discriminant, take the square root, and only round to the required accuracy at the very last step.
- Confusing equations and identities in "show that" questions. If you are asked to show that an expression simplifies to a given result, you must work from one side to the other, not write the answer and work backwards.
Self-check questions
- y is directly proportional to x3. When x = 2, y = 40. Find y when x = 3. (Answer: k = 5, y = 5 x 27 = 135)
- Solve simultaneously: 2x + y = 7 and x - y = 2. (Answer: x = 3, y = 1)
- Solve x2 + 2x - 15 = 0. (Answer: (x + 5)(x - 3) = 0, x = -5 or x = 3)
- Solve 3x2 - 5x - 1 = 0 using the quadratic formula. Give answers to 2 d.p. (Answer: x = 1.85 or x = -0.18)
- Solve the inequality 4x + 3 <= 19. (Answer: x <= 4)
- Find the integer values of n that satisfy -3 < 2n - 1 <= 7. (Answer: n = 0, 1, 2, 3, 4)
The equations, formulae and identities: proportion to inequalities edexcel igcse topics covered here are tested on both papers, and these edexcel igcse mathematics specification a explained methods connect directly to coordinate geometry, graphing, and problem-solving questions across the specification. For more igcse 4ma1 equations, formulae and identities: proportion to inequalities practice, work through edexcel igcse mathematics specification a practice questions on the Green Bridge CBT platform to identify gaps and build confidence before exam day.
Edexcel IGCSE Mathematics Specification A revision notes on proportion, simultaneous equations, quadratic equations and inequalities with worked examples.
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