🔢 Quant

Linear Equations

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

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~4%

of section questions

Est. hours⏱️

~10h

to master

~1/22

2021

~2/22

2022

~1/22

2023

~2/22

2024

1/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~0.9 per slot (2021: 0.7 · 2022: 1.3 · 2023: 0.3 · 2024: 1.3 · 2025: 1.0). Linear equations and systems are the steady word-problem workhorse of CAT algebra — almost always dressed as an application (ages, prices, portfolios, mixtures in disguise). The heavier root-and-discriminant work is split out under Quadratic Equations (~1.7 per slot, the bigger block).

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

Name the unknowns, turn each clause into one equation, then combine — don't over-solve. : three equal-cost baskets give two equations in the apple/orange/mango prices; solve in terms of M (A = 2M, 3M = 4O) to rewrite any basket as a multiple of M and read off the equivalent mangoes. : the target 2a + 3b − 3c is a fixed linear blend of the two given equations, so hunt the multipliers instead of solving for a, b and c separately. : two lines share no point exactly when the coefficient ratios match but the constant ratio differs — the parallel-lines test, not brute force. Translate, then exploit the structure.

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

27 practice questions

Easy

For any non-zero real number x, f(x) + 4f(1/x) = 5x. Then, the sum of all possible values of x for which f(x) = -8 is

Easy

A shop wants to sell a certain quantity (in kg) of lentils. It sells half the quantity and an additional 12 kg of these lentils to the first customer. Then, it sells half of the remaining quantity and an additional 12 kg to the second customer. Then, it sells half of the remaining quantity and an additional 12 kg to the third customer. Finally, when the shop sells half of the remaining quantity and an additional 12 kg to the fourth customer, there are no lentils left. The initial quantity, in kg, of lentils is

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2021 · Slot 1

Medium

A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, or a basket of 8 oranges and 7 mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

CAT 2025 · Slot 1

Medium

If a − 6b + 6c = 4 and 6a + 3b − 3c = 50, where a, b and c are real numbers, the value of 2a + 3b − 3c is

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