Quantitative Aptitude · Equations

Simultaneous Linear Equations for CA Foundation

Updated 1 October 2026 · Fact-checked

Simultaneous linear equations are two or more linear equations that share the same unknowns and must all hold together. You solve them by substitution, elimination or cross-multiplication. In exams, eliminate one variable first, find the other, then put both values back into the original equations to check.

Understand Simultaneous Linear Equations

A linear equation in two unknowns, such as 2x + 3y = 12, has endless solutions. Each value of x gives a matching y. When a second equation joins it, only the pair (x, y) that satisfies both is accepted. That pair is the solution of the simultaneous equations.

Graphically, each equation is a straight line. The solution is the point where the lines cross. Two lines can cross at one point, never cross (parallel), or lie on top of each other. That gives three cases: one solution, no solution, or infinitely many.

Every method does the same job: reduce the system to one equation in one unknown. Substitution writes one variable in terms of the other and puts it into the second equation. Elimination adds or subtracts the equations so one variable cancels. Cross-multiplication is a ready formula built from the coefficients.

For three unknowns, use elimination twice. Remove the same variable from two different pairs of equations. This gives two equations in two unknowns. Solve those, then go back to find the third variable.

Key formulas to remember

Standard form (two unknowns): a₁x + b₁y = c₁ and a₂x + b₂y = c₂; Bring both equations to this form before using any method. Keep signs with the coefficients.
Cross-multiplication result: x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁); Valid only when a₁b₂ − a₂b₁ ≠ 0. The denominator is the same for both x and y.
Unique solution test: a₁ ÷ a₂ ≠ b₁ ÷ b₂; The lines cross at exactly one point. Use cross-multiplication or elimination.
No solution test: a₁ ÷ a₂ = b₁ ÷ b₂ ≠ c₁ ÷ c₂; The lines are parallel. The system is inconsistent.
Infinite solutions test: a₁ ÷ a₂ = b₁ ÷ b₂ = c₁ ÷ c₂; Both equations represent the same line. The system is consistent but dependent.
Three unknowns strategy: Eliminate one variable from two pairs of equations, then solve the resulting 2 × 2 system; Pick the variable that is easiest to cancel. Find the third variable by back-substitution.

How to solve Simultaneous Linear Equations questions

This method works for any question on simultaneous linear equations, including word problems.

1Write every equation in standard form with variables on the left and the constant on the right. Clear fractions and brackets first.
2Choose the variable to eliminate. Prefer one whose coefficients are equal, opposite, or easy to match.
3Multiply one or both equations by suitable numbers so the chosen variable has the same coefficient in both.
4Add the equations if the signs are opposite. Subtract them if the signs are the same. This leaves one equation in one unknown.
5Solve that equation. Put the value into any original equation to find the next unknown.
6For three unknowns, repeat the elimination on a different pair so you get a second equation in the same two unknowns. Solve these two, then find the third.
7Check the answer in an original equation, preferably one you did not use for back-substitution.
8Read what the question asks. It may want x + y, xy, or 2x − y, not just x.

Quickest way: Option-check and add-subtract shortcuts

When to use it: Use in the MCQ paper when each option gives a value or a set of values, or when the coefficients are small.

Look for a quick combination. Adding or subtracting the two equations often gives x + y or x − y directly, which may be all the question asks.
If options list (x, y) pairs, test an option in the simpler equation first. Reject any pair that fails.
Test the second equation only for options that survive the first. Usually one option remains.
For three unknowns, adding all three equations may give x + y + z at once. Subtracting from this gives each variable quickly.
Use the cross-multiplication formula when coefficients are awkward, such as 7 and 13, and the denominator is easy to compute.
If a question needs more than about two minutes, mark it and move on. Wrong answers cost 0.25 marks each.

Common mistakes in Simultaneous Linear Equations

Sign errors when subtracting equations
Students subtract only the first term and forget to change the sign of the rest of the second equation.
Fix: Write the second equation with all signs reversed on a new line, then add. Or always eliminate by adding after multiplying by a negative number.
Multiplying only the left side when scaling an equation
Rushing leaves the constant unchanged.
Fix: When you multiply an equation by k, multiply every term, including the constant on the right.
Using the cross-multiplication formula with equations not in standard form
Students read c as a coefficient from an equation like 2x + 3y − 12 = 0 and keep the wrong sign.
Fix: First rewrite as 2x + 3y = 12. Then c is 12, not −12.
Stopping after finding one variable
Students are relieved to get x and forget y or the combination asked for.
Fix: Underline what the question asks before solving. Check that your final value matches it.
Not checking for inconsistent or dependent systems
Students assume every system has one solution.
Fix: Compare a₁ ÷ a₂, b₁ ÷ b₂ and c₁ ÷ c₂ first. If the first two ratios are equal, the system has no unique solution.
Eliminating different variables inconsistently in a three-variable system
Students remove x from one pair and y from another, so the two new equations have different unknowns.
Fix: Choose one variable and eliminate that same variable from two different pairs.

Worked examples

Example 1

If 3x + 2y = 18 and 5x − y = 17, then the value of 2x − y is: (A) 3 (B) 5 (C) 8 (D) 11

Show the solution

Eliminate y. Multiply the second equation by 2: 10x − 2y = 34.
Add this to the first equation: 3x + 2y + 10x − 2y = 18 + 34, so 13x = 52 and x = 4.
Put x = 4 in 5x − y = 17: 20 − y = 17, so y = 3.
Check in the first equation: 3(4) + 2(3) = 12 + 6 = 18. Correct.
Compute 2x − y = 8 − 3 = 5.

Answer: (B) 5

Example 2

If x + y + z = 6, x − y + z = 2 and 2x + y − z = 1, then the value of xyz is: (A) 4 (B) 6 (C) 8 (D) 12

Show the solution

Subtract the second equation from the first: (x + y + z) − (x − y + z) = 6 − 2, so 2y = 4 and y = 2.
Put y = 2 in the first equation: x + z = 4.
Put y = 2 in the third equation: 2x + 2 − z = 1, so 2x − z = −1.
Add x + z = 4 and 2x − z = −1: 3x = 3, so x = 1. Then z = 3.
Check the second equation: 1 − 2 + 3 = 2. Correct.
Compute xyz = 1 × 2 × 3 = 6.

Answer: (B) 6

Example 3

3 pens and 2 notebooks cost ₹130. 2 pens and 5 notebooks cost ₹215. The total cost of 1 pen and 1 notebook is: (A) ₹45 (B) ₹55 (C) ₹60 (D) ₹65

Show the solution

Let a pen cost ₹p and a notebook cost ₹n. Then 3p + 2n = 130 and 2p + 5n = 215.
To eliminate p, multiply the first by 2 and the second by 3: 6p + 4n = 260 and 6p + 15n = 645.
Subtract the first from the second: 11n = 385, so n = 35.
Put n = 35 in 3p + 2n = 130: 3p + 70 = 130, so p = 20.
Check the second equation: 2(20) + 5(35) = 40 + 175 = 215. Correct.
Cost of 1 pen and 1 notebook = 20 + 35 = ₹55.

Answer: (B) ₹55

Exam tips

Read the last line of the question first. Often you need x + y or a similar combination, which you can get by adding the equations.
In MCQs, substitute options into the simplest equation to cut down choices quickly.
For word problems, define the variables on paper in one line. Most errors come from setting up the wrong equation, not from solving it.
Questions on consistency ask about parallel or coincident lines. Learn the ratio tests, since they take seconds to apply.
Always check your answer in one original equation. It takes ten seconds and saves 1 mark plus the 0.25 penalty.

Practice questions from Equations

Simultaneous Linear Equations: frequently asked questions

What is the difference between substitution and elimination method?

Substitution expresses one variable in terms of the other and puts it into the second equation. Elimination adds or subtracts the equations to cancel a variable. Elimination is usually faster when coefficients are whole numbers. Substitution suits cases where one variable already has coefficient 1.

When should I use the cross-multiplication method?

Use it when coefficients are awkward and you want to avoid several steps of scaling. Write the equations in standard form, apply the formula, and check that the denominator a₁b₂ − a₂b₁ is not zero. Be careful with signs.

How do I solve three variable simultaneous equations in CA Foundation?

Eliminate the same variable from two different pairs of equations. This gives two equations in two unknowns. Solve them, then put the values into an original equation to find the third variable.

What if the equations have no solution or infinite solutions?

If a₁ ÷ a₂ = b₁ ÷ b₂ but this differs from c₁ ÷ c₂, the lines are parallel and there is no solution. If all three ratios are equal, the equations are the same line and have infinite solutions. Otherwise there is one unique solution.