Introduction to algebra
Section: Algebra and graphs 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Letters as Generalised Numbers, and Substitution
A letter in algebra stands in for a number whose value can change or is not yet known. Substitution means replacing each letter with a given number and working out the result, following the normal order of operations.
- A letter (such as x, a or n) represents a generalised number - the same expression works for any value substituted in
- To substitute, replace every occurrence of the letter with its given value, then evaluate using BIDMAS
- It often helps to put the substituted value in brackets first, especially when it is negative
- A formula relates several letters together (e.g. A = (1/2)(a+b)h) - substituting values for every letter except one allows that one to be found
Worked Example: Substituting into an Expression
- Question: Find the value of 3x + 5 when x = 4.
- Step 1: Replace x with 4: 3(4) + 5
- Step 2: Evaluate: 12 + 5 = 17
- Answer: 17
Worked Example: Substituting into a Formula
- Question: The formula for the area of a trapezium is A = (1/2)(a+b)h. Find the value of A when a = 6, b = 10 and h = 4.
- Step 1: Substitute each value: A = (1/2)(6+10)(4)
- Step 2: Work through BIDMAS: A = (1/2)(16)(4) = (1/2)(64)
- Answer: A = 32
Worked Example: Substituting a Negative Number
- Question: Find the value of x² - 3x when x = -2.
- Step 1: Replace x with (-2), keeping the brackets: (-2)^2 - 3(-2)
- Step 2: Evaluate each term: (-2)^2 = 4, 3 × (-2) = -6
- Step 3: Combine: 4 - (-6) = 4 + 6 = 10
- Answer: 10
Common Mistakes
MistakeLosing a negative sign when substituting a negative value, e.g. finding x² when x = -2 as -4 instead of 4
Fixput the substituted value in brackets before evaluating: (-2)² = (-2) × (-2) = 4, since a negative times a negative is positive
MistakeApplying the wrong order of operations after substituting, e.g. treating 3x + 5 as 3(x + 5)
Fixsubstitution does not change the structure of the expression - only the multiplication term (3 × x) happens before the addition, exactly as in the original expression
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