CAT Quant Linear Equations: Methods and Solved Examples

Linear equations are an important part of CAT Quant, especially in age, cost, ratio, and other word-based problems. Understanding how to form equations correctly makes these questions easier to solve.

Linear Equations cover single-variable equations, substitution and elimination methods, and step-by-step solutions to common CAT-style problems. It also explains how to handle age and cost problems, verify answers, and prepare effectively for Linear Equations in CAT Quant.

What Is a Linear Equation in CAT Quant?

Linear equations are one of the simpler topics in CAT Algebra. They still appear in many forms inside word problems, so the basic idea must be clear first.

A linear equation is an equation in which the highest power of the variable is 1. It can have one variable, two variables or even three variables. For example, 3x + 5 = 20 has one variable, and 2x − y = 15 has two. As long as no variable is squared or cubed, the equation is linear.

This topic sits under Algebra in CAT Quantitative Aptitude. You can also check the full CAT syllabus to see where Algebra fits.

How to Solve a Single-Variable Linear Equation?

A single-variable equation is the easiest type. The goal is to get the variable alone on one side.

• Move all constants to the other side. A plus sign becomes a minus, and a minus becomes a plus.

• Divide both sides by the coefficient of the variable.

Take 3x + 5 = 20. Moving 5 gives 3x = 15, and dividing by 3 gives x = 5.

How Does the Method of Substitution Work?

When there are two variables, you need two equations. These are called simultaneous linear equations. In substitution, you write one variable in terms of the other and put it into the second equation.

Take 2x − y = 15 and 3x + 2y = 27.

1. From the first equation, y = 2x − 15.

2. Put this in the second equation: 3x + 2(2x − 15) = 27.

3. Simplify: 3x + 4x − 30 = 27, so 7x = 57 and x = 57/7.

4. Put x back in y = 2x − 15. This gives y = 114/7 − 105/7 = 9/7.

You can isolate either variable first. Choose the form that makes the calculation easier. Check the answer by putting both values in the original equations. You should get 15 and 27.

How Does the Method of Elimination Work?

In elimination, you remove one variable completely and solve for the other. First, make the coefficients of the chosen variable equal. Then add or subtract the equations.

The sign rule is simple. Look only at the coefficients of the variable you want to remove, and ignore the other variable.

Use the same example. The y coefficients are −1 and +2. Multiply the first equation by 2 to get 4x − 2y = 30. Now the y terms have opposite signs, so add this to 3x + 2y = 27. The y terms cancel, and you get 7x = 57. This gives the same x = 57/7 as before.

Remember to multiply every term of the equation, including the right-hand side. Both methods give the same answer, so use the one you find easier.

How to Solve Word Problems on Linear Equations?

In CAT, solving the equation is easy. The hard part is reading the question and forming the equation correctly.

Follow these steps every time:

1. Read the question and understand the language.

2. Define the variables.

3. Write the equations.

4. Choose substitution or elimination.

5. Solve the equations.

6. Check the answer against the original conditions.

How to Solve Age Problems Using Linear Equations?

In age problems, one person's age is taken as a variable, and the other person's age is written in terms of it. Time words need care.

"After n years" or "n years hence" means every age increases by n. "n years ago" means every age decreases by n. Always count these from the present, and keep the time period the same before you compare two ages.

Example 1: A father's age is six times his son's age. After 4 years, the father's age will be four times the son's age. Let the son's age be x, so the father's age is 6x. After 4 years, the equation is 6x + 4 = 4(x + 4). This gives 2x = 12, so x = 6. The son is 6 years old, and the father is 36.

Example 2: John's present age is one-fourth of his father's age 2 years ago. His father's age after 10 years will be twice Raman's age after 10 years. Raman celebrated his 12th birthday 2 years ago.

• Let the father's age 2 years ago be 4x. Then John's age now is x.

• The father's present age is 4x + 2, and after 10 years it will be 4x + 12.

• Raman is 14 now and will be 24 after 10 years.

• So 4x + 12 = 2 × 24 = 48, which gives x = 9.

Taking the father's age as 4x avoids fractions. This is a useful trick whenever one age is a fraction of another.

How to Solve Quantity and Cost Problems?

These questions use two variables, such as two types of tea at different prices. One equation comes from the total quantity, and the other comes from the total cost.

A shop sells blend A at Rs. 100 per kg and blend B at Rs. 150 per kg. A customer buys 10 kg in total and pays Rs. 1300. Let the quantities be A kg and B kg.

• Quantity equation: A + B = 10

• Cost equation: 100A + 150B = 1300

Divide the second equation by 50 to get 2A + 3B = 26. Put B = 10 − A to get 2A + 30 − 3A = 26, so A = 4 and B = 6.

There is a shortcut. Since the total is known, write one quantity as A and the other as 10 − A. This needs only one variable.

How to Solve Ratio-Based Purchase Problems?

This was the "impact" question in the session. The quantities are unknown, but the ratio is given, and the costs are unknown positive whole numbers.

Ashok buys pens and pencils from the same shop twice. In the first visit, the ratio is 2:3, and he pays Rs. 86. In the second visit, the ratio is 4:1, and he buys 4 pens for a total of Rs. 112. Let a pen cost Rs. x and a pencil cost Rs. y.

A ratio of 2:3 can mean 2 and 3, or 4 and 6, or 20 and 30. In the second visit, 4 pens and 1 pencil cost Rs. 112. If the first visit had 4 pens and 6 pencils, it would cost more than Rs. 112, but he paid only Rs. 86. So the first visit must be 2 pens and 3 pencils.

• Visit 1: 2x + 3y = 86

• Visit 2: 4x + y = 112

Substituting y = 112 − 4x gives 2x + 336 − 12x = 86, so x = 25 and y = 12. The amount paid for pens in the second visit is 4 × 25 = Rs. 100.

Tips to Prepare Linear Equations for CAT

Build accuracy in CAT Linear Equations by practising equation formation, choosing efficient solving methods, handling TITA questions, and applying concepts through regular mock tests.

• Practise forming equations from the language of the question, since solving is the easy part.

• Do not depend only on options. CAT also has TITA questions, where you type the answer and no options are given.

• Learn both methods and pick the one you are more comfortable with.

• Always count "before" and "after" from the present time.

• Take regular mock tests. You can also check the CAT 2026 exam page for updates.

CAT Quant Linear Equations become easier once you are comfortable with equation formation, substitution and elimination. Focus on understanding the language of word problems, especially questions based on ages, costs and ratios, and always verify your final answer.

Regular practice with CAT-level questions and mock tests can improve both accuracy and speed. Students can also explore PW’s CAT preparation resources for structured practice and doubt support.