Introduction to Algebra
Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core
1. Algebraic notation
Letters standing for numbers
- Algebra uses letters in place of numbers that are unknown or that stand for any value
- A letter used this way is called a variable
- Knowing what a letter represents matters as much as the working, so questions define it before using it
Writing operations
- Addition and subtraction are written with the usual signs
- Multiplication has no sign at all: the letters are simply written together
- ab means a × b, and 5ab means 5 × a × b
- A number is written before the letters, so 5a rather than a5
- Division is written as a fraction rather than with a division sign
- a/b means a ÷ b
- Powers and roots behave exactly as they do with numbers
- a² means a × a, and 4a² squares first and multiplies by 4 afterwards
- Brackets group in the usual way, so 3(a + b) adds first and multiplies the result by 3
- The order of operations is unchanged, which is what settles how an expression such as ab + c/3 is read
2. The vocabulary
Terms
- A term is a single part of an expression: a number, a letter, or numbers and letters multiplied together
- 20, x, 5x, 6xy and 4x² are all terms
- A term that is a number alone is a constant, because its value never changes
- The number multiplying the letters is the coefficient
- In 8m the coefficient is 8, and in −t it is −1, since a coefficient of 1 is not written
Expressions, equations, formulas and identities
| Name | What it is |
|---|---|
| Expression | Terms combined by operations, with no equals sign and nothing to solve |
| Equation | Two expressions set equal, true only for particular values |
| Formula | A rule connecting quantities, used to calculate one from the others |
| Identity | A statement true for every value, written with ≡ |
Factors
- A factor of a term divides it exactly
- 5 and x are both factors of 5x, because 5 × x = 5x
- A common factor divides two or more terms, and the highest one is what factorising takes out
- 12ab and 18b share the factor 6b, which is the highest available
3. Collecting like terms
Which terms combine
- Like terms have exactly the same letters raised to exactly the same powers, whatever their coefficients
- 2x and 3x are like, and so are 5xy and −7xy
- Different letters or different powers make terms unlike, so they cannot be combined
- 2x and 3y are unlike, and so are 4x² and 6x⁴
- Order of multiplication does not matter, so xy and yx count as like terms
Combining them
- Add or subtract the coefficients and keep the letters unchanged
- Each term carries the sign written in front of it, and that sign travels with the term if the order is rearranged
- A subtraction can be read as adding a negative, which keeps the signs straight when terms are reordered
Worked example
Simplifying by collecting like terms
Simplify 7p + 4q − 2p − 9q.
Solution:
- Like terms carry exactly the same letter to exactly the same power, and only like terms can be combined
- Here there are two families, the p terms and the q terms
- Take the sign in front of each term with it, so the four terms are +7p, +4q, −2p and −9q
- The p terms: 7p − 2p = 5p
- The q terms: 4q − 9q = −5q, which is negative because 9 is the larger number
- The simplified expression is 5p − 5q
- The two parts cannot be combined any further, because p and q stand for different quantities, so 5p − 5q does not cancel to 0
- Check by substituting: p = 2 and q = 1 gives 14 + 4 − 4 − 9 = 5 in the original, and 10 − 5 = 5 in the answer
4. Substitution
Replacing letters with numbers
- Substitution means putting given numbers in place of the letters, then evaluating
- Write the expression or formula down first, then replace each letter, then calculate
- Follow the order of operations exactly as with any numerical calculation
Negative values need brackets
- A negative value substituted for a letter is written in brackets, which keeps the sign attached through powers and multiplications
- Where x = −4, x² is (−4)² = 16, not −16
- Without brackets a calculator applies the power before the sign, giving the wrong result
Worked example
Substituting a negative value
Given a = 5 and b = −3, find the value of (i) 3a − b² (ii) 2b(a + b).
Solution:
- Substitute a negative value inside brackets every time, so the sign cannot be separated from the number
- (i) The expression becomes 3 × 5 − (−3)²
- BIDMAS puts the index before the subtraction, so square first
- (−3)² = (−3) × (−3) = 9, positive because two negatives multiply to a positive
- 15 − 9 = 6
- Without the bracket this reads 3 × 5 − −3², and since the power is applied before the sign, −3² is −9 and the answer comes out as 15 + 9 = 24, which is wrong
- (ii) The expression becomes 2 × (−3) × (5 + (−3))
- Brackets first: 5 + (−3) is 5 − 3 = 2
- Then left to right: 2 × (−3) = −6, and −6 × 2 = −12
- The answer is negative because exactly one of the factors is negative
Exam tip
Write the substituted line first
Substituting into a formula was set in 8 of the 22 papers. Almost every scheme gives the first mark for the line with the numbers in place, such as 3 × 5 − (−3)² above or 58 = 2v² − (−8) below. So write that line down before working anything out, rather than doing it in your head: if the arithmetic after it slips, the line still keeps one mark.
Worked example
Finding a letter that is squared in a formula
In the formula E = 2v² − u, find the positive value of v when E = 58 and u = −8.
Solution:
- Substitute the values you know, with the negative in brackets: 58 = 2v² − (−8)
- Subtracting −8 is the same as adding 8, so 58 = 2v² + 8
- Take 8 from both sides: 2v² = 50, then divide by 2: v² = 25
- The formula gives v², so finish with a square root: v = √25 = 5
- Check: 2 × 5² − (−8) = 50 + 8 = 58, which is the value of E given