Ch 3 · 2.5 Equations · IGCSE Mathematics 0580
Topic 3 of 9 in Algebra and Graphs — IGCSE Mathematics 0580 — 49 lessons.
Introduction
An equation is a balance: whatever you do to one side, do to the other. This lesson builds up from simple linear equations to fractions, pairs of simultaneous equations and quadratics, and finally rearranging formulas. Many exam questions start by asking you to form the equation from words or a diagram. (Syllabus section 2.5.)
Key ideas — Forming and solving linear equations
Translate words: "the product of two consecutive even numbers" is 2n(2n + 2). Solve by expanding brackets, collecting the unknown on one side, and dividing: 5 - 2x = 3(x + 7) ⇒ 5 - 2x = 3x + 21 ⇒ -16 = 5x ⇒ x = -3.2.
🧑🏫 Teacher's explanation
Think of an equation like a balanced seesaw. The equals sign (=) is the pivot point. Whatever you do to one side—adding weight, removing weight, or doubling it—you must do exactly the same to the other side to keep the seesaw perfectly level. If you add 5 to the left, you must add 5 to the right.
📖 Definitions
🔍 Worked example — Easy
Solve the linear equation: 5 - 2x = 3(x + 7)
🔍 Worked example — Standard
Solve the fractional equation: x/2 + x+1/4 = 3
🔍 Worked example — Exam-style
Solve the simultaneous equations: \begin{cases} 3x + 2y = 16 \\ 5x - 2y = 0 \end{cases}
⚠️ Common errors
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🎯 Exam question
Key ideas — Fractional equations
Multiply every term by the common denominator to clear the fractions:
🧑🏫 Teacher's explanation
Imagine you are trying to share pizzas, but some pizzas are cut into x+2 slices and others into 2x-1 slices. It is very difficult to add them together when the 'size' of the slices is different. To make it easy, we want to turn everything into 'whole pizzas' by multiplying everything by a common denominator.
📖 Definitions
🔍 Worked example — Easy
Solve x/3 + x/4 = 5
🔍 Worked example — Standard
Solve 2/x+2 + 3/2x-1 = 1
🔍 Worked example — Exam-style
Solve x+1/x-1 = 4/x
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Simultaneous linear equations
Elimination: make one coefficient match, then add or subtract. For 3x + 2y = 12 and 5x - 3y = 1: × 3 and × 2 give 9x + 6y = 36 and 10x - 6y = 2; adding, 19x = 38, x = 2, then y = 3. Substitution is better when one equation gives a letter directly (y = dots).
Key ideas — Quadratic equations
Rearrange to ax^2 + bx + c = 0 first. Then:
🧑🏫 Teacher's explanation
A quadratic equation is like a puzzle where the unknown value is squared, meaning there are often two different numbers that can make the equation true. Think of it like a symmetric valley or a hill; the solutions are the two points where the curve hits the ground (the x-axis).\n\nTo solve these, you first need to get everything on one side so the equation equals zero, looking like ax^2 + bx + c = 0. Once it looks like that, you have three main tools: factorising (splitting it into two brackets), completing the square (turning it into a perfect square plus/minus a constant), or using the 'magic' quadratic formula which works every single time, even when the numbers are messy.
📖 Definitions
🔍 Worked example — Easy
Solve x^2 - 5x + 6 = 0 by factorisation.
🔍 Worked example — Standard
Solve x^2 + 8x + 10 = 0 by completing the square. Give your answer in surd form.
🔍 Worked example — Exam-style
Solve 3x^2 - 7x + 2 = 0 using the quadratic formula. Give your answer to 2 decimal places.
⚠️ Common errors
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🎯 Exam question
Key ideas — Changing the subject
Undo operations in reverse order. If the subject appears twice, collect those terms and factorise:
🧑🏫 Teacher's explanation
Changing the subject of a formula is like unwrapping a gift. If you want to get to the gift (the subject), you have to remove the wrapping paper, the ribbon, and the box in the reverse order they were put on. In algebra, 'wrapping' means operations like addition, multiplication, or powers. To 'unwrap' them, we use inverse operations: subtraction undoes addition, division undoes multiplication, and squaring undoes a square root.
📖 Definitions
🔍 Worked example — Easy
Make x the subject of 3x - 5 = 11.
🔍 Worked example — Standard
Make a the subject of v = u + at.
🔍 Worked example — Exam-style
Make x the subject of y = x + 5/x - 2.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples
A rectangle is 3 cm longer than it is wide and its area is 40 cm². x(x + 3) = 40 ⇒ x^2 + 3x - 40 = 0 ⇒ (x + 8)(x - 5) = 0. Width 5 cm (reject -8), length 8 cm.
🔍 Worked example — Easy
Solve the fractional equation: x/2x + 1 = 4
🔍 Worked example — Standard
A rectangle has a length that is 3 cm longer than its width. Given the area is 40 cm², find the width x.
🔍 Worked example — Exam-style
Make r the subject of the formula: V = 1/3π r^2h
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 3x - 1 = 2x + 10 ⇒ x = 11. (2) y = 4x - 7: 2x + 12x - 21 = 7 ⇒ x = 2, y = 1. (3) x = -4 ± √(40)/6: 0.39 or -1.72. (4) ax - cx = d + ab ⇒ x = d + ab/a - c. (5) n^2 + n - 30 = 0 ⇒ n = 5 or n = -6.