🔢 Quant

Linear Equations

Single and simultaneous linear equations and their word-problem applications.

12%

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~12%

of section questions

Est. hours⏱️

~10h

to master

~2/22

2021

~3/22

2022

3/22

2023

~3/22

2024

~4/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~2.6 per slot (2021: 1.3 · 2022: 2.7 · 2023: 3.0 · 2024: 2.3 · 2025: 3.7). Equation-solving is the biggest single block in CAT Quant — and growing: from ~2 to ~3 per slot across 2021–2025.

Linear Equations

Equations where every variable is to the first power. CAT rarely asks you to "just solve" — it dresses them as word problems, or probes how many solutions a system has.

How many solutions?

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

Condition	Solutions	Lines
$\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}$	exactly one	intersecting
$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}$	none	parallel
$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$	infinite	same line

A worked example

$2x+3y=12$ and $x+y=5$ . Find $x$ and $y$ .

Substitute $x=5-y$ from the second equation:

$2(5-y)+3y=12\ \Rightarrow\ 10+y=12\ \Rightarrow\ y=2,\ x=3.$

Elimination is often faster: scale the second to $2x+2y=10$ and subtract from the first to get $y=2$ in one step.

Word-problem discipline

Name the unknowns explicitly ("let the son's age be $s$ ").
Turn each sentence into one equation; count that you have as many equations as unknowns.
For "two-digit number" problems, a number with digits $t,u$ is $10t+u$ — reversing gives $10u+t$ .

🎯PYQ Evidence

Translate the words into relations, then exploit the structure — equate, factor, or square. : three baskets costing the same give two equations in the apple/orange/mango prices; solving them in terms of M (here A = 2M and 3M = 4O) lets you rewrite any basket's cost as a multiple of M, which directly reads off the equivalent number of mangoes. : pull out the common variable to get y(x+z) = 19 and z(x+y) = 51; since 19 is prime and x, y, z are natural, y must be 1, collapsing the rest to a single quadratic and the minimum xyz = 34. : move everything to one side and regroup into (x+2y)² + (x−2y−1)² = 0; a sum of squares is zero only when each is zero, so x − 2y = 1 falls out instantly. Read the words as equations, then let primality or a perfect square do the collapsing.

Common traps

Hidden dependence. Two equations that are multiples of each other give infinitely many solutions, not one.
Integer constraints. "Number of people/coins" must be a non-negative integer — sometimes that alone pins the answer.

Checklist

Define variables in words first
One equation per independent condition
Use the ratio test to decide one / none / infinite solutions
Prefer elimination when a variable cancels cleanly

Sample Questions

29 practice questions

Easy

A political candidate collected $1,749 from a fundraising dinner. If each supporter contributed at least $50, what is the greatest possible number of contributors at the dinner?

Easy

Janet is now 25 years younger than her mother Carol. If in 6 years Janet's age will be half Carol's age, how old was Janet 5 years ago?

Create a free account to access all 29 practice questions on this topic.

CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 1

TITAMedium

The number of non-negative integer values of k for which the quadratic equation $x^{2}$ − 5x + k = 0 has only integer roots, is

Your answer

CAT 2024 · Slot 1

Easy

A shop wants to sell a certain quantity (in kg) of grains. It sells half the quantity and an additional 3 kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 3 kg to the second customer. Finally, when the shop sells half of the remaining quantity and an additional 3 kg to the third customer, there are no grains left. The initial quantity, in kg, of grains is

Create a free account to access all 8 CAT PYQs on this topic.

Continue Your Prep

Flashcards →

Bite-size concept cards

PYQ Practice →

Filter from 1,002 PYQs

Mock Test →

Full CAT simulation

Practice Linear Equations