Study Notes: Equations
Quantitative Ability – XAT
Advanced University‑Level (MIT‑style) Treatment
Prerequisites
- Basic algebra (variables, operations, identities)
- Understanding of functions and graphs
- Elementary number theory (divisibility, modular arithmetic)
Goal
Equip you with a rigorous, first‑principles understanding of the most common equation types encountered in XAT, including edge cases and advanced concepts that sharpen problem‑solving intuition.
1. Introduction
| Category | Typical XAT Questions | Typical Difficulty |
|---|---|---|
| Linear | One‑variable, two‑variable, systems | Easy – 1–2 steps |
| Quadratic | Roots, discriminants, factorisation | Medium – 3–4 steps |
| Higher‑degree | Cubic, quartic, rational roots | Hard – 5+ steps |
| Functional | Find f such that f(x+1)=… | Very hard – 6–8 steps |
| Inequalities | Solve for x in ax²+bx+c>0 | Medium – 3–5 steps |
| Diophantine | Integer solutions | Hard – 4–6 steps |
Equations are the backbone of quantitative reasoning. Mastering them gives you a systematic toolkit for almost all other topics.
2. Fundamental Algebraic Elements
| Symbol | Meaning | Key Properties |
|---|---|---|
| x, y, z, … | Variables – unknowns that can take any real/complex value | Commutative: a+b=b+a, a·b=b·a |
| c, k, … | Constants – fixed numbers | Associative: (a+b)+c=a+(b+c) |
| + – · – ÷ – ^ | Operations | Distributive: a(b+c)=ab+ac |
Rule of Thumb: Always isolate the variable before performing non‑linear operations (e.g., square roots, logarithms).
3. Linear Equations
3.1 One‑Variable
Standard Form:
[
ax + b = 0,\quad a\neq 0
]
Solution:
[
x = -\frac{b}{a}
]
Edge Cases
- (a=0) & (b=0): Infinite solutions.
- (a=0) & (b\neq 0): No solution.
3.2 Two‑Variable
Standard Form:
[
ax + by = c
]
Solution Methods
- Substitution
[ x = \frac{c - by}{a} ] - Elimination
Multiply equations to cancel one variable.
Edge Cases
- Parallel lines (no intersection).
- Coincident lines (infinitely many solutions).
3.3 Systems of N Equations
Matrix Representation
[
A\mathbf{x} = \mathbf{b}
]
- Unique Solution: (\det(A)\neq 0).
- No Solution: Rank(A) ≠ Rank([A|b]).
- Infinite Solutions: Rank(A) = Rank([A|b]) < n.
Gaussian Elimination – step‑by‑step reduction to echelon form.
Caution: Zero pivot → swap rows; keep track of row operations to avoid fraction loss.