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Linear Equation Questions for CAT 2025
Linear Equations is an important topic in the CAT Quant section. For CAT 2025, you can expect questions that test both your speed and accuracy. These questions often involve solving equations, understanding word problems, and applying concepts in areas like ratios, ages, and work. Sometimes, they also include extra conditions or tricky cases.
In this blog, you’ll find the most important types of CAT linear equation questions, important formulas, common mistakes to avoid, and practice problems to help you improve and score better.
Important Formulas for CAT Linear Equations Questions
If you're preparing for Linear Equations questions in CAT 2025, knowing a few key formulas can really help you solve problems faster and more easily. We’ve also shared a PDF with all the important formulas, so you can download it and revise anytime.
Download CAT Linear Equations Formula PDF
| Category | Formula / Concept | Explanation / When to Use |
|---|---|---|
| Basic Linear Equation (1 variable) | ax+b=0⇒x=−baax + b = 0 \Rightarrow x = -\frac{b}{a}ax+b=0⇒x=−ab | Use for single-variable equations. |
| System of 2 Linear Equations (Substitution) | Solve one equation for one variable and plug into the other. | Use when one variable is easily isolated. |
| System of 2 Linear Equations (Elimination) | Multiply equations to align coefficients and eliminate one variable. | Use when variables can be canceled out easily. |
| System of 2 Linear Equations (Cramer’s Rule) | x=∣c1b1c2b2∣∣a1b1a2b2∣,y=∣a1c1a2c2∣∣a1b1a2b2∣x = \frac{\begin{vmatrix}c_1 & b_1 \\ c_2 & b_2\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix}},\quad y = \frac{\begin{vmatrix}a_1 & c_1 \\ a_2 & c_2\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix}}x=a1a2b1b2c1c2b1b2,y=a1a2b1b2a1a2c1c2 | Only when determinant ≠0\neq 0=0. Clean coefficients preferred. |
| System of 3 Equations (Elimination/Substitution) | Use pairwise elimination to reduce to 2 variables, then solve. | Use when a clean system is given. |
| Consistency Check | If a1a2=b1b2≠c1c2⇒\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \Rightarrowa2a1=b2b1=c2c1⇒ No solution. Infinite if all 3 ratios equal. | Determines if a system is consistent. |
| Word Problem (Sum) | "Sum of two numbers is S" → x+y=Sx + y = Sx+y=S | Used in number, age problems. |
| "One number is r times another" | y=r⋅xy = r \cdot xy=r⋅x | Used in age, work, ratio problems. |
| Age Problems | "n years ago" → x−nx - nx−n, "in n years" → x+nx + nx+n | Always define present age as variable. |
| Time-Work Problems | Rate = 1time\frac{1}{\text{time}}time1, Combined rate = Sum of individual rates | Use when dealing with work efficiency. |
| Mixture/Average Problems | Weighted Avg.=ax+byx+y\text{Weighted Avg.} = \frac{a x + b y}{x + y}Weighted Avg.=x+yax+by | Use for mixing quantities of different costs. |
| Speed-Distance-Time | Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}Distance=Speed×Time | Use in linear motion questions. |
| Linear Inequalities | ax+b<c⇒x<c−baax + b < c \Rightarrow x < \frac{c-b}{a}ax+b<c⇒x<ac−b | Solve similarly to linear equations. Reverse sign for negative aaa. |
| Graph of Linear Equation | y=mx+cy = mx + cy=mx+c: straight line with slope mmm, intercept ccc | Helpful for understanding relationships visually. |
| Two-variable linear equations in geometry | Parallel lines → same slope (a1b1=a2b2)\left(\frac{a_1}{b_1} = \frac{a_2}{b_2}\right)(b1a1=b2a2), no solution unless same line. | Useful in set theory, coordinate geometry cases. |
| Number Properties | Integers, even/odd, constraints → handle carefully in equations | Especially in number-based word problems. |
Common Mistakes To Avoid in Linear Equations Questions
When practicing Linear Equations for CAT, many students make small but common mistakes. Knowing these in advance can help you avoid losing marks:
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Misreading the question: Words like “more than,” “less than,” “times,” or “is” can be tricky. Read carefully to write the correct equation.
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Forgetting limits on variables: For example, ages can’t be negative, and denominators in equations shouldn’t be zero.
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Calculation mistakes: Small sign errors while adding, subtracting, or eliminating variables can change the whole answer. Always double-check your steps.
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Missing special cases: Some systems of equations may have no solution or many solutions. Look out for these cases.
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Not checking your answer: After solving, always plug the values back into the original question to make sure they work.
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Avoiding fractions: Don’t ignore answers just because they’re fractions—unless the question says only whole numbers are allowed.
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Wasting too much time on tough equations: If a problem looks complicated, look for patterns or shortcuts instead of doing long calculations.
List Of CAT Linear Equation Questions
Question 1
In NutsAndBolts factory, one machine produces only nuts at the rate of 100 nuts per minute and needs to be cleaned for 5 minutes after production of every 1000 nuts.
Another machine produces only bolts at the rate of 75 bolts per minute and needs to be cleaned for 10 minutes after production of every 1500 bolts. If both the machines start production at the same time, what is the minimum duration required for producing 9000 pairs of nuts and bolts?
correct answer:- 3
Question 2
Mayank, Mirza, Little and Jaspal bought a motorbike for $60. Mayank paid one-half of the sum of the amounts paid by the other boys. Mirza paid one-third of the sum of the amounts paid by the other boys. Little paid one-fourth of the sum of the amounts paid by the other boys. How much did Jaspal have to pay?
correct answer:- 2
Question 3
A change-making machine contains one-rupee, two-rupee and five-rupee coins. The total number of coins is 300. The amount is Rs. 960. If the numbers of one-rupee coins and two-rupee coins are interchanged, the value comes down by Rs. 40. The total number of five-rupee coins is
correct answer:- 2
An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs 1200 and Rs 2400 respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs 5400.
Question 4
What is the weight of Praja’s luggage?
correct answer:- 4
Question 5
My son adores chocolates. He likes biscuits. But he hates apples. I told him that he can buy as many chocolates he wishes. But then he must have biscuits twice the number of chocolates and should have apples more than biscuits and chocolates together. Each chocolate cost Re 1. The cost of apple is twice the chocolate and four biscuits are worth one apple. Then which of the following can be the amount that I spent on that evening on my son if number of chocolates, biscuits and apples brought were all integers?
correct answer:- 1
Question 6
A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is
correct answer:- 2
Question 7
The last time Rahul bought Diwali cards, he found that the four types of cards that he liked were priced Rs.2.00, Rs.3.50, Rs.4.50 and Rs.5.00 each. As Rahul wanted 30 cards, he took five each of two kinds and ten each of the other two, putting down the exact number of 10 rupees notes on the counter payment. How many notes did Rahul give?
correct answer:- 4
Question 8
The sum of two integers is 10 and the sum of their reciprocals is 5/12. Then the larger of these integers is
correct answer:- 3
Question 9
In Sivakasi, each boy's quota of match sticks to fill into boxes is not more than 200 per session. If he reduces the number of sticks per box by 25, he can fill 3 more boxes with the total number of sticks assigned to him. Which of the following is the possible number of sticks assigned to each boy?
Click For More CAT Linear Equations questions
correct answer:- 2
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