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Quantitative Aptitude · Equations

Linear Equations in One Variable for CA Foundation

Updated 1 October 2026 · Fact-checked

A linear equation in one variable has the form ax + b = 0, where a ≠ 0, and has exactly one solution, x = −b ÷ a. To solve it, collect variable terms on one side, constants on the other, then divide by the coefficient. For word problems, define x, translate each sentence into an equation, solve, and check.

Understand Linear Equations in One Variable

An equation is a statement that two expressions are equal. A linear equation in one variable has only one unknown (say x), and the highest power of that unknown is 1. Examples: 3x + 5 = 20 and 2(x − 4) = x + 7.

Solving means finding the value of x that makes both sides equal. That value is the root or solution. The idea is simple: an equation is like a balance. Whatever you do to one side, you must do to the other. Add, subtract, multiply or divide both sides by the same number (not zero) and the equation stays true.

Every linear equation in one variable can be reduced to ax + b = 0 with a ≠ 0. This gives exactly one root, x = −b ÷ a. If the x terms cancel out and you are left with a false statement like 3 = 5, there is no solution. If you are left with a true statement like 4 = 4, every value of x works. These two cases are rare in MCQs but possible.

Most marks come from word problems. The work is translation: pick the unknown, turn each sentence into a mathematical statement, and solve. Age problems, number problems and money problems all follow this pattern. Equations with fractions are handled by multiplying both sides by the LCM of the denominators.

Key formulas to remember

Standard form
ax + b = 0, a ≠ 0
Every linear equation in one variable reduces to this form.
Solution
x = −b ÷ a
Gives the single root. It needs a ≠ 0.
Balance rule
Do the same operation on both sides
Never multiply or divide by zero. Dividing by an expression that can be zero may lose or add roots.
Consecutive numbers
x, x + 1, x + 2 (integers); x, x + 2, x + 4 (odd or even)
Use x − 1, x, x + 1 or x − 2, x, x + 2 when the sum or middle value is asked.
Digits of a two-digit number
Number = 10t + u
t is the tens digit and u is the units digit. The reversed number is 10u + t.
Age after or before n years
Age now ± n
Add n for the future, subtract n for the past. Apply it to every person in the problem.
Money value
Total value = number of items × value of each item
Keep the units (₹ or paise) the same on both sides.

How to solve Linear Equations in One Variable questions

Use this method for every equation or word problem on the topic. It keeps your working short and catches errors early.

  1. 1Read the question once fully and mark what is asked. Decide which quantity will be x. Usually it is the quantity you are asked to find.
  2. 2Write every other quantity in terms of x using the facts given, such as 'three times', 'more than' or 'after 5 years'.
  3. 3Form one equation from the sentence that links the quantities, such as a total, a sum or a ratio.
  4. 4Remove brackets and clear fractions by multiplying both sides by the LCM of the denominators.
  5. 5Collect the x terms on one side and the constants on the other. Change the sign when a term crosses the equals sign.
  6. 6Divide both sides by the coefficient of x to get the value.
  7. 7Check the value in the original words of the question, not only in your equation.
  8. 8Make sure you answer what was asked. The question may want the largest number or the age after some years, not x itself.

Quickest way: Option substitution and smart setup

When to use it: Use it in MCQs when the equation is awkward or the setup is slow. Word problems with whole-number answers suit it best.

  1. Read the four options. If the equation is long, test the options instead of solving.
  2. Start with the middle-sized option. Substitute and see whether the condition holds. If the result is too high or too low, you know which direction to go.
  3. For age problems, test the options against the 'after n years' condition first, since it is the hardest to satisfy by chance.
  4. For numbers and sums, use divisibility or parity to cross out options. For example, three consecutive numbers with sum 105 means the middle is 105 ÷ 3.
  5. Choose the variable that makes the equation simplest. The middle term for consecutive numbers is one example.
  6. If a problem needs more than about 90 seconds, mark it and move on. Wrong answers cost 0.25 marks, so guess only when you have removed at least one option.

Common mistakes in Linear Equations in One Variable

  • Not changing the sign when moving a term across the equals sign.

    Students move terms by habit and forget that they are really adding or subtracting on both sides.

    Fix: Write the operation on both sides, for example '− 5 on both sides', for the first few practice questions. Then it becomes automatic.

  • Multiplying only the first term inside a bracket, such as 2(x − 4) = 2x − 4.

    Rushing through expansion.

    Fix: Multiply every term inside the bracket. 2(x − 4) = 2x − 8. Draw arrows from the outside number to each term.

  • Clearing fractions but not multiplying every term, including whole numbers.

    Students focus on the fractions and leave the plain terms unchanged.

    Fix: Multiply each term on both sides by the LCM. In x/2 + 3 = x/3, multiplying by 6 gives 3x + 18 = 2x.

  • Giving x as the answer when the question asks for a different quantity.

    Solving feels like the finish line.

    Fix: Underline the final ask in the question. After solving, compute the asked value, for example the larger number or the father's age after 10 years.

  • Applying 'after n years' to only one person in an age problem.

    The sentence feels like it refers to one person, but the time passes for everyone.

    Fix: Add n to every age involved. Write both 'now' and 'after' columns before forming the equation.

  • Mixing units in money problems, such as rupees and paise.

    Values are given in different forms in the same question.

    Fix: Convert everything to one unit before forming the equation.

Worked examples

Example 1

A father is three times as old as his son. After 12 years, the father will be twice as old as the son. What is the present age of the son? (A) 10 years (B) 12 years (C) 15 years (D) 18 years

Show the solution
  1. Let the son's present age be x years. The father's present age is 3x years.
  2. After 12 years, the son will be x + 12 and the father will be 3x + 12.
  3. The condition gives 3x + 12 = 2(x + 12).
  4. Expand: 3x + 12 = 2x + 24.
  5. Collect x terms: 3x − 2x = 24 − 12, so x = 12.
  6. Check: the father is 36 now. After 12 years, the ages are 48 and 24, and 48 = 2 × 24. This is correct.

Answer: (B) 12 years

Example 2

The sum of three consecutive odd numbers is 105. What is the largest of the three numbers? (A) 33 (B) 35 (C) 37 (D) 39

Show the solution
  1. Let the middle odd number be x. The three numbers are x − 2, x and x + 2.
  2. Their sum is (x − 2) + x + (x + 2) = 3x.
  3. So 3x = 105, which gives x = 35.
  4. The numbers are 33, 35 and 37. Their sum is 105.
  5. The question asks for the largest number, which is x + 2 = 37.

Answer: (C) 37

Example 3

A cashier has 70 notes of ₹100 and ₹50 that total ₹5,000. How many ₹100 notes does the cashier have? (A) 20 (B) 30 (C) 40 (D) 50

Show the solution
  1. Let the number of ₹100 notes be x. The number of ₹50 notes is 70 − x.
  2. The total value is 100x + 50(70 − x) = 5,000.
  3. Expand: 100x + 3,500 − 50x = 5,000.
  4. Simplify: 50x = 1,500, so x = 30.
  5. Check: 30 notes of ₹100 give ₹3,000. 40 notes of ₹50 give ₹2,000. The total is ₹5,000 and the note count is 70.

Answer: (B) 30

Exam tips

  • In age problems, build a small table with the columns 'now' and 'after/before'. This avoids the commonest error of changing only one age.
  • If the options are whole numbers and the equation is messy, substitute the options. Start with a middle value to save time.
  • Always read the last line of the question again. Many options are traps that give x or a related value rather than the quantity asked.
  • For equations with fractions, multiply by the LCM first. Do not add fractions one by one.
  • Treat a question as a skip if the setup is not clear within 30 seconds. Negative marking of 0.25 per wrong answer makes blind guessing costly.

Practice questions from Equations

Linear Equations in One Variable: frequently asked questions

What is a linear equation in one variable?

It is an equation with one unknown whose highest power is 1, such as 3x + 7 = 22. It can always be written as ax + b = 0 with a ≠ 0. It has exactly one solution, x = −b ÷ a.

How do I solve word problems on linear equations quickly?

Let x be the quantity asked for, write the other quantities in terms of x, and form one equation from the main condition. Solve it, then check it against the question. In MCQs, testing the options is often faster.

How do I solve age problems in CA Foundation?

Write each person's present age in terms of x. Then add or subtract the number of years from every person's age to express the future or past condition. Equate the two sides as the question says, solve, and check with the actual ages.

Can a linear equation in one variable have no solution or infinite solutions?

Yes, in special cases. If the variable cancels and you get a false statement like 2 = 5, there is no solution. If you get a true statement like 3 = 3, every value of x works.