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Linear Equations for CAT: Formulas, Weightage & Question Types

Words into equations, integer solutions, and how many solutions a system has.

Linear Equations formula sheet

Everything CAT needs from Linear Equations, in one place. Bookmark it for revision; the lesson in the app explains where each line comes from and when to reach for it.

Everything for this chapter, in one place.

Two equations, two unknowns — how many solutions

For a1x+b1y=c1a_1x + b_1y = c_1 and a2x+b2y=c2a_2x + b_2y = c_2:

Exactly one solutiona1a2≠b1b2\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}
No solutiona1a2=b1b2≠c1c2\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} — parallel lines
Infinitely manya1a2=b1b2=c1c2\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} — the same line twice
Quick test for uniquenessa1b2−a2b1≠0a_1b_2 - a_2b_1 \neq 0

Integer solutions of ax+by=cax + by = c

Solutions existonly if gcd⁡(a,b)\gcd(a,b) divides cc
Once one solution is foundthe rest step by x→x+bgx \to x + \dfrac{b}{g}, y→y−agy \to y - \dfrac{a}{g}, with g=gcd⁡(a,b)g=\gcd(a,b)
Parity shortcutif aa is even and cc is odd, then byby must be odd
Counting themfind the allowed range for one variable, then count the steps inside it

Under-determined systems

Fewer equations than unknownsyou cannot find the variables — but a combination of them is often fixed
Solook for the combination the question asks for, and build it from the equations
With integer constraintsthe gap is closed by divisibility, not by algebra
With an ordering constraintthe gap is closed by an inequality, giving a maximum or minimum

Standard word-problem shapes

Fixed cost plus per-unit costtotal=F+vn\text{total} = F + vn — two data points give FF and vv
Per-unit profit against total profitmultiply by nn before comparing
"Half the remainder and 3 more", repeatedwork backwards from the end

The 4 Linear Equations question types CAT repeats

Past CAT papers return to the same few shapes. Learn to recognise them and most questions become a known routine.

  1. 01

    Integers, bounds, and a maximum

  2. 02

    A word problem that becomes two equations

  3. 03

    More unknowns than equations

  4. 04

    How many solutions does this system have

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Linear Equations in CAT: quick answers

How many Linear Equations questions come in CAT Quant?
About 0.8 questions in a 22-question Quant section, worth roughly 2 marks. Linear Equations appeared in 12 of the 24 CAT papers from 2017 to 2025, 16 questions in all, which ranks it 16 of 25 Quant chapters.
Are Linear Equations questions in CAT MCQ or TITA?
44% of the Linear Equations questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.
What types of Linear Equations questions does CAT ask?
CAT keeps returning to 4 question shapes in Linear Equations: Integers, bounds, and a maximum; A word problem that becomes two equations; More unknowns than equations; How many solutions does this system have.

More Algebra for CAT