Algebra
Solving linear equations and simultaneous equations.
This page is a sequence, not a document: try a question cold, learn the methods, drill until they’re automatic, then prove it under a clock. Skipping straight to reading is the least effective thing you can do here — that’s not opinion, it’s the testing effect.
Try one first
Before any teaching — a real question tells you honestly where you stand.
Solve for x: 3x + 14 = 29
The methods
Every algebra pattern the tests use — the method in a few lines, then a worked example.
Linear equation
Undo the constant first (add or subtract it), then divide by the coefficient.
Worked example
Solve for x: 5x − 7 = 28
Answer: 7
Add 7: 5x = 35. Divide by 5: x = 7. The 4.2 foil divides 28 by 5 and ignores the −7; 35 forgets to divide; 21 just computes 28−7.
Substitution
Substitute the value carefully, squaring only what's squared, then evaluate term by term.
Worked example
If y = 3x² − 4x + 2, what is y when x = 5?
Answer: 57
Substitute x=5: 3×25 − 4×5 + 2 = 75 − 20 + 2 = 57. The 207 foil squares (3×5) instead of just x; 97 adds the −4x term; 53 subtracts the +2.
Rearranging formulas
Rearrange to isolate the unknown before substituting the numbers.
Worked example
Given v = u + at, make a the subject and find a when v = 30, u = 6, t = 4.
Answer: 6
Rearrange: a = (v − u)/t = (30 − 6)/4 = 24/4 = 6. The 9 foil adds u instead of subtracting; 24 forgets to divide by t.
Simultaneous equations
Add or subtract the equations to eliminate one unknown, solve, then back-substitute.
Worked example
Solve the pair and give x: 2x + y = 11 and x − y = 1
Answer: 4
Add the equations to eliminate y: 3x = 12, so x = 4 (and y = 3). The 3 foil gives y; 12 forgets to divide by 3.
Inequalities
Solve the inequality, then pick the smallest or largest whole number that satisfies it strictly.
Worked example
For the smallest whole number n, 3n + 4 is greater than 25. What is n?
Answer: 8
3n + 4 > 25 means 3n > 21, so n > 7. The smallest whole number greater than 7 is 8. The 7 foil includes the boundary (3·7+4 = 25, which is not greater than 25).
Word problems
Translate the sentence into an equation (total = fixed + rate × quantity), then solve step by step.
Worked example
A technician charges a £45 call-out fee plus £30 for each hour worked. A repair bill came to £195. How many hours did the technician work?
Answer: 5
Translate the words: total = fee + rate × hours, so 195 = 45 + 30h. Subtract the fee: 30h = 150. Divide by the rate: h = 5 hours. The 6.5 foil divides the whole bill by £30 and forgets the £45 fee. The 2.6 foil divides by fee and rate added together. The 5.5 foil subtracts one hour's rate (£30) instead of the £45 fee.
Age problems
Set the unknown as x, express the other quantities in terms of x, build the equation from the condition, and solve.
Worked example
A father is four times as old as his son. In five years, he will be three times as old. How old is the son now?
Answer: 10
Let the son be x, the father 4x. In five years: 4x + 5 = 3(x + 5) → 4x + 5 = 3x + 15 → x = 10. The 40 foil gives the father's age now; 45 the father in five years; 15 the son in five years.
Drill it
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Prove it under a clock
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