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Lesson 14 min· Section 1 of 4

The straight-up lesson, built for the exam.

Linear Equations

Formula sheet

Everything for this chapter, in one place.

Two equations, two unknowns — how many solutions

For $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ :


Exactly one solution	$\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$
No solution	$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ — parallel lines
Infinitely many	$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ — the same line twice
Quick test for uniqueness	$a_1b_2 - a_2b_1 \neq 0$

Integer solutions of $ax + by = c$


Solutions exist	only if $\gcd(a,b)$ divides $c$
Once one solution is found	the rest step by $x \to x + \dfrac{b}{g}$ , $y \to y - \dfrac{a}{g}$ , with $g=\gcd(a,b)$
Parity shortcut	if $a$ is even and $c$ is odd, then $by$ must be odd
Counting them	find the allowed range for one variable, then count the steps inside it

Under-determined systems


Fewer equations than unknowns	you cannot find the variables — but a combination of them is often fixed
So	look for the combination the question asks for, and build it from the equations
With integer constraints	the gap is closed by divisibility, not by algebra
With an ordering constraint	the gap is closed by an inequality, giving a maximum or minimum

Standard word-problem shapes


Fixed cost plus per-unit cost	$\text{total} = F + vn$ — two data points give $F$ and $v$
Per-unit profit against total profit	multiply by $n$ before comparing
"Half the remainder and 3 more", repeated	work backwards from the end

Why this chapter matters

0.8

questions in a typical Quant section

16th

of 25 chapters

≈ 2 of 66 marks a section ride on it.

Asked in12 of 24 papers

20172025

Year by yearper section

2017: 02017: 0 per section
2018: 02018: 0 per section
2019: 02019: 0 per section
2020: 1.12020: 1.1 per section
2021: 12021: 1 per section
2022: 0.72022: 0.7 per section
2023: 0.72023: 0.7 per section
2024: 0.72024: 0.7 per section
2025: 12025: 1 per section

Question type16 asked

MCQ 56%Type-in 44%

Type-in answers carry no negative marking.

From every Linear Equations question in CAT papers since 2017.