Algebra basics
Algebra basics · linear equations · quadratic equations
10 min read · Updated 2026-05-26
What is Algebra?
Algebra is the practice of naming an unknown quantity with a letter and turning a worded condition into an equation that can be solved. One linear equation settles one unknown and two unknowns need two equations; a quadratic is cracked by factorisation or by x = (−b ± √(b² − 4ac))/2a, whose roots sum to −b/a and multiply to c/a.
The story
"A father is 4 times as old as his son. After 20 years, he will be only twice as old. Find their present ages."
Brute-force approach: Try father = 30, son = 7.5 → no. Try 40, 10 → after 20 yrs, 60 vs 30. Yes! Coincidence? Or did you guess?
Algebraic approach (10 seconds): Let son = x. Father = 4x. After 20: father = 4x + 20; son = x + 20. 4x + 20 = 2(x + 20) → 4x + 20 = 2x + 40 → 2x = 20 → x = 10.
Algebra doesn't require luck. Every age problem, every "some-thing-times-something" word problem, becomes a one-equation, one-unknown affair.
CSAT loves age problems, sum-difference problems, and linear-equation-disguised-as-puzzle problems. After this unit, every such question is a 60-second template.
Why this matters for UPSC
For CSAT (Paper II, qualifying 33%):
- 6-9 questions per paper are pure algebra (linear or quadratic).
- Algebra is the universal solver — when number-system tricks fail, when alligation gets messy, set variables and write equations.
- Quadratic equations appear in 1-2 CSAT questions per year (max/min problems, "find a number such that…").
- Real-life: Budget planning, age/year calculations, monthly EMI computations.
The single most transferable maths skill for CSAT. Master linear equations and 70% of the qualifying paper becomes mechanical.