Algebra

What you'll learn

This revision guide covers all algebraic topics tested in WJEC GCSE Mathematics, from Foundation to Higher tier. You'll learn to manipulate algebraic expressions, solve equations and inequalities, work with sequences, understand functions and interpret graphs. These skills form approximately 30% of your GCSE examination and provide essential tools for problem-solving across mathematics.

Key terms and definitions

Coefficient — the numerical factor multiplying a variable (in 5x, the coefficient is 5)

Expression — a mathematical phrase combining numbers, variables and operations without an equals sign (e.g., 3x + 7)

Linear equation — an equation where the highest power of the variable is 1, forming a straight line when graphed

Quadratic expression — an expression containing a term in x² as the highest power (general form: ax² + bx + c)

Factorising — writing an expression as a product of its factors, the reverse process of expanding brackets

Simultaneous equations — two or more equations with multiple unknowns that must be solved together to find values satisfying all equations

Gradient — the steepness of a line, calculated as change in y ÷ change in x, often denoted as m

Inequality — a mathematical statement using symbols <, >, ≤ or ≥ to compare expressions

Core concepts

Simplifying and manipulating expressions

Collecting like terms involves combining terms with identical variable parts. Only terms with exactly the same letter combinations can be added or subtracted:

• 5x + 3x = 8x
• 7a² - 2a² = 5a²
• 4xy + 2x - xy + 5 = 3xy + 2x + 5 (note that xy and x are different)

Multiplying terms requires multiplying coefficients and adding powers of the same base:

• 3x × 4y = 12xy
• 2a² × 5a³ = 10a⁵
• -3p × 4p = -12p²

Expanding brackets (also called multiplying out) means removing brackets by multiplication:

• Single brackets: 5(2x - 3) = 10x - 15
• Double brackets: (x + 4)(x + 3) = x² + 3x + 4x + 12 = x² + 7x + 12
• Use FOIL or grid method: First, Outer, Inner, Last terms

Factorising reverses expansion. For single brackets, extract the highest common factor (HCF):

• 6x + 9 = 3(2x + 3)
• 4x² - 8x = 4x(x - 2)

For quadratic expressions (Higher tier), find two numbers that multiply to give the constant term and add to give the coefficient of x:

• x² + 7x + 12 = (x + 3)(x + 4)
• x² - 5x + 6 = (x - 2)(x - 3)

Difference of two squares uses the identity a² - b² = (a + b)(a - b):

• x² - 16 = (x + 4)(x - 4)
• 9x² - 25 = (3x + 5)(3x - 5)

Solving equations and inequalities

Linear equations require isolating the variable through inverse operations:

1. Eliminate fractions by multiplying throughout
2. Expand any brackets
3. Collect variable terms on one side, numbers on the other
4. Simplify and divide by the coefficient

Example approach for 3(x - 2) = 15:

• Expand: 3x - 6 = 15
• Add 6: 3x = 21
• Divide by 3: x = 7

Equations with unknowns on both sides need variables collected on one side:

For 5x + 3 = 2x + 15:

• Subtract 2x from both sides: 3x + 3 = 15
• Subtract 3: 3x = 12
• Divide by 3: x = 4

Quadratic equations (Higher tier) can be solved by:

Factorising method:

1. Rearrange to standard form (= 0)
2. Factorise
3. Set each factor equal to zero
4. Solve for x

Quadratic formula when factorising is difficult: x = [-b ± √(b² - 4ac)] / 2a for ax² + bx + c = 0

Completing the square for (x + p)² + q form

Simultaneous equations have two main solution methods:

Elimination method:

1. Make coefficients of one variable equal (multiply if needed)
2. Add or subtract equations to eliminate that variable
3. Solve for remaining variable
4. Substitute back to find the other

Substitution method:

1. Rearrange one equation for one variable
2. Substitute into the other equation
3. Solve and substitute back

Inequalities follow the same rules as equations except:

• Reverse the inequality sign when multiplying or dividing by a negative number
• Solutions are represented on number lines using open circles (< or >) and filled circles (≤ or ≥)

Sequences and nth term

Arithmetic sequences have a constant difference between consecutive terms.

The nth term formula is: Un = a + (n - 1)d

Where:

• a = first term
• d = common difference
• n = term position

For sequence 5, 8, 11, 14...:

• First term a = 5
• Common difference d = 3
• nth term: Un = 5 + (n - 1) × 3 = 3n + 2

Geometric sequences (Higher tier) multiply by a constant ratio each time:

• 2, 6, 18, 54... has common ratio 3
• nth term: Un = ar^(n-1) where r is the common ratio

Quadratic sequences have a second difference that is constant:

• 3, 7, 13, 21, 31... (first differences: 4, 6, 8, 10; second difference: 2)
• nth term contains n²: general form Un = an² + bn + c

Functions and graphs

Function notation uses f(x) to represent "function of x":

• f(x) = 3x + 2 means the output is 3 times the input plus 2
• f(5) means substitute x = 5: f(5) = 3(5) + 2 = 17

Linear graphs have equation y = mx + c where:

• m = gradient (steepness)
• c = y-intercept (where line crosses y-axis)

Finding gradient from two points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁)/(x₂ - x₁)

Parallel lines have equal gradients.

Perpendicular lines have gradients whose product is -1 (e.g., m₁ = 2 and m₂ = -½)

Quadratic graphs (Higher tier) have shape y = ax² + bx + c:

• Form U-shaped parabolas
• a > 0 gives U shape; a < 0 gives ∩ shape
• Turning point (vertex) is maximum or minimum
• Roots are x-intercepts where y = 0

Other graphs to recognise (Higher tier):

• Cubic: y = x³ (S-shaped curve)
• Reciprocal: y = 1/x (two separate curves)
• Exponential: y = aˣ (rapid growth/decay)

Changing the subject of formulae

Rearranging formulae uses the same inverse operations as solving equations:

To make r the subject of A = πr²:

• Divide by π: A/π = r²
• Square root: r = √(A/π)

For formulae with the subject appearing twice (Higher tier):

1. Expand brackets if needed
2. Collect all terms containing the subject on one side
3. Factorise to extract the subject
4. Divide by remaining bracket

To make x the subject of y = (ax + b)/(cx + d):

• Multiply both sides by (cx + d): y(cx + d) = ax + b
• Expand: cxy + dy = ax + b
• Collect x terms: cxy - ax = b - dy
• Factorise: x(cy - a) = b - dy
• Divide: x = (b - dy)/(cy - a)

Algebraic fractions

Simplifying algebraic fractions requires factorising numerator and denominator:

(x² - 9)/(x + 3) = (x + 3)(x - 3)/(x + 3) = x - 3

Adding and subtracting needs a common denominator:

3/x + 2/(x + 1) = [3(x + 1) + 2x]/[x(x + 1)] = (5x + 3)/[x(x + 1)]

Multiplying fractions: multiply numerators and denominators, then simplify

Dividing fractions: multiply by the reciprocal of the divisor

Worked examples

Example 1: Solving simultaneous equations (Foundation/Higher, 4 marks)

Question: Solve the simultaneous equations:

• 3x + 2y = 16
• 5x - 2y = 8

Solution:

Step 1: Notice the y coefficients are +2 and -2 (ready to eliminate). [1 mark for method]

Step 2: Add equations together: (3x + 2y) + (5x - 2y) = 16 + 8 8x = 24

Step 3: Solve for x: [1 mark] x = 3

Step 4: Substitute x = 3 into first equation: 3(3) + 2y = 16 9 + 2y = 16 2y = 7 y = 3.5 [1 mark]

Step 5: Check in second equation: 5(3) - 2(3.5) = 15 - 7 = 8 ✓

Answer: x = 3, y = 3.5 [1 mark for both correct]

Example 2: Factorising and solving a quadratic (Higher tier, 4 marks)

Question: Solve x² - 3x - 18 = 0 by factorising.

Solution:

Step 1: Find two numbers that multiply to -18 and add to -3. Factors of -18: (1, -18), (-1, 18), (2, -9), (-2, 9), (3, -6), (-3, 6) Check: -6 + 3 = -3 ✓ and -6 × 3 = -18 ✓ [1 mark]

Step 2: Write factorised form: (x - 6)(x + 3) = 0 [1 mark]

Step 3: Set each bracket equal to zero: x - 6 = 0 or x + 3 = 0 [1 mark]

Step 4: Solve: x = 6 or x = -3 [1 mark]

Example 3: Finding nth term of a sequence (Foundation/Higher, 3 marks)

Question: Find the nth term of the sequence 7, 10, 13, 16, 19...

Solution:

Step 1: Find the common difference: 10 - 7 = 3, 13 - 10 = 3 (constant difference = 3) [1 mark]

Step 2: Write general form for arithmetic sequence: nth term = dn + c, where d = common difference nth term = 3n + c [1 mark]

Step 3: Find c by substituting n = 1: When n = 1: 3(1) + c = 7 c = 4

Answer: nth term = 3n + 4 [1 mark]

Check: When n = 2: 3(2) + 4 = 10 ✓

Common mistakes and how to avoid them

• Sign errors when expanding brackets: Remember -3(x - 5) = -3x + 15, not -3x - 15. Multiply the negative sign through every term inside the bracket.

• Forgetting both solutions for quadratics: When you have (x - 3)(x + 2) = 0, both x = 3 AND x = -2 are solutions. Set each bracket equal to zero separately.

• Mishandling inequality signs: When multiplying or dividing by a negative number, reverse the inequality. For -2x > 6, dividing by -2 gives x < -3, not x > -3.

• Confusing coefficient and exponent operations: 2x × 3x = 6x², not 6x or 5x². Multiply coefficients (2 × 3 = 6) and add powers of x (x¹ × x¹ = x²).

• Incorrect substitution into formulae: For f(x) = x² - 3x when x = -2, write f(-2) = (-2)² - 3(-2) with brackets to avoid sign errors. Answer: 4 + 6 = 10.

• Adding unlike terms: You cannot simplify 3x + 2y or x² + x. Only combine terms with identical variable parts.

Exam technique for Algebra

• "Solve" means find specific values: Show all working steps. For equations worth 3+ marks, you'll lose marks for answer-only, even if correct.

• "Simplify" or "expand and simplify": Must collect like terms in your final answer. Leaving 5x + 3x uncombined loses the final mark.

• Show factorising clearly: Write the factorised form explicitly before solving. In x² - 5x + 6 = 0, write (x - 2)(x - 3) = 0 as a separate line worth its own mark.

• Mark allocation guides working: A 4-mark question needs 4 steps of working. A 1-mark question might be answer-only, but showing method protects against calculation errors earning you method marks.

Quick revision summary

Algebra involves manipulating expressions, solving equations and working with sequences and graphs. Master expanding and factorising, including quadratics on Higher tier. Linear equations require inverse operations; quadratics need factorising or the formula. Simultaneous equations use elimination or substitution. Arithmetic sequences follow Un = a + (n-1)d. Linear graphs use y = mx + c; know how to find gradients. Higher tier includes completing the square, algebraic fractions, and various graph types. Always show working methodically for maximum marks.