Elimination Method for Simultaneous Equations

Elimination Method for Simultaneous Equations solves two linear equations by adding or subtracting suitable multiples of them to remove one variable. The resulting single-variable equation gives the first value, which is then substituted into either original equation to find the second value. This page covers formulas, steps, examples and common cases.

On this page

What Is the Elimination Method for Simultaneous Equations?

The elimination method solves simultaneous linear equations by cancelling one variable through addition or subtraction of the equations. After finding one variable, substitute its value into an original equation to calculate the other.

For equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, first make the coefficients of x or y equal in magnitude. Add the equations if the selected coefficients have opposite signs, and subtract them if they have the same sign. For example, 2x + y = 7 and x − y = 2 give 3x = 9 on addition, so x = 3 and y = 1.

Elimination Method for Simultaneous Equations Video Lessons

Watch short topic-wise lessons for quick revision.

12 Lessons
Lesson 1 of 12 Quick Revision

Simultaneous Equations Elimination Method

Learn how to solve simultaneous linear equations by eliminating one variable, simplifying the resulting equation, and finding the values of both unknowns.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Elimination Method for Simultaneous Equations Lessons Scroll to explore →

Elimination Method for Simultaneous Equations Formula & Tricks

Important Formulas

General elimination formula for x
x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁)

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, this formula gives x when a₁b₂ − a₂b₁ ≠ 0.

General elimination formula for y
y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁)

Use this formula for y under the same non-zero denominator condition.

Eliminating y by addition
If b₁ = −b₂, then (a₁ + a₂)x = c₁ + c₂

Add the equations when the coefficients of y are equal in magnitude but have opposite signs.

Eliminating y by subtraction
If b₁ = b₂, then (a₁ − a₂)x = c₁ − c₂

Subtract one equation from the other when the coefficients of y are equal with the same sign.

Quick Tricks

Choose the variable needing the smallest multiplier

Compare the coefficients of x and y. Eliminate the variable whose coefficients can be made equal with smaller or simpler multipliers.

Example: For 3x + 2y = 12 and 5x + 4y = 24, multiply the first equation by 2 and eliminate y directly.
Track the sign before adding or subtracting

Opposite signs require addition, while identical signs require subtraction. Write the multiplied equations separately before combining them.

Example: From 2x + 3y = 13 and 4x − 3y = 5, add the equations to get 6x = 18, so x = 3.
Check both values in an original equation

Substitution into either original equation verifies arithmetic and signs without repeating the full elimination process.

Example: For x = 3 and y = 1 in 2x + y = 7, the left side is 2(3) + 1 = 7, which matches the right side.

Elimination Method for Simultaneous Equations Concepts

Equalising Coefficients Before Elimination

Multiply one or both equations by suitable constants so that one variable has equal coefficients in magnitude.

For 2x + 3y = 13 and 3x + 2y = 12, the y-coefficients can be made equal by multiplying the first equation by 2 and the second by 3. This gives 4x + 6y = 26 and 9x + 6y = 36. Subtracting produces 5x = 10, so x = 2. Substitution gives y = 3.

Example: 2(2x + 3y = 13): 4x + 6y = 26; 3(3x + 2y = 12): 9x + 6y = 36; subtraction gives x = 2 and y = 3.

Adding or Subtracting the Equations

Add equations when the selected variable has opposite coefficients; subtract them when its coefficients have the same sign.

For 2x + y = 7 and x − y = 2, the y terms cancel on addition: 3x = 9, so x = 3. For 2x + 3y = 11 and 2x − y = 3, subtracting the second equation from the first cancels 2x and gives 4y = 8, so y = 2.

Example: (2x + 3y) − (2x − y) = 11 − 3 gives 4y = 8, hence y = 2.

Finding the Remaining Variable by Substitution

After eliminating one variable, substitute the value found into either original equation to determine the other variable.

If elimination gives x = 4 and the original equation is 2x + y = 11, then 2(4) + y = 11, so y = 3. Using an original equation avoids carrying any multiplication errors from the elimination step.

Example: For 3x + 2y = 18 and x − 2y = 2, addition gives 4x = 20, so x = 5. Substituting in x − 2y = 2 gives y = 3/2.

Dependent and Inconsistent Equations

If elimination produces 0 = 0, the equations have infinitely many solutions; if it produces a false statement such as 0 = 5, they have no solution.

Equations 2x + 4y = 8 and x + 2y = 4 are equivalent, so elimination gives 0 = 0 and infinitely many solutions. Equations 2x + 4y = 8 and x + 2y = 5 have proportional left sides but different constants; elimination gives 0 = −2, so no solution.

Example: Multiplying x + 2y = 4 by 2 gives the first equation, showing that both equations represent the same line.

Learn more concepts on PrepShots

Start ₹1 Trial →

Practice Elimination Method for Simultaneous Equations Questions

Practise published questions related to this topic.

1(√3 + √2) + 1/((√3 - √2)) = ?→ 2Which of the following pairs of numbers are co-prime?→ 3The profit earned on selling an article for Rs. 2000 is 19 times of the loss incurred on selling the same article for Rs. 1200. What is the profit percentage on this article at the selling price of Rs. 2000?→ 4If 4 lines of a print, in a column of a book, contain 63 words, how many words will a column of 64 lines contain?→ 5The listed price of a book is ₹800. Find the final selling price of the book if successive discounts of 10%, 20% and 25% are given on the list price.→ 6An agent sells a pressure cooker with a marked price of ₹8,460 at a discount of 30%. What is the selling price (in ₹) of the pressure cooker?→ 7What is the largest 2 digit number which is exactly divisible by both 2 and 7?→ 8If 3/7 of 70 percent of a number is 900, then what is the number?→ 9Find the number of pairs of coprime numbers between 11 and 19 (both inclusive).→ 10A trader lost 10% by selling a shirt for ₹900. If he gains x% by selling it for ₹1,450, the value of x is:→

Want more practice questions?

Practice in App →

Elimination Method for Simultaneous Equations Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Elimination Method Revision Points

Use these rules to revise the standard procedure and special cases.

  • Write both equations in a consistent form, preferably ax + by = c.
  • Make the coefficients of one variable equal in magnitude.
  • Add opposite-sign terms and subtract same-sign terms.
  • Find one variable, then substitute it into an original equation.
  • If elimination gives 0 = 0, there are infinitely many solutions.
  • If elimination gives a false statement such as 0 = 5, there is no solution.
  • For a unique solution, the determinant a₁b₂ − a₂b₁ must be non-zero.

Related Topic Explainers

Substitution Method for Simultaneous Equations → Cross Multiplication Method for Linear Equations → Solving Quadratic Equations by Factorisation → Quadratic Formula → Discriminant of a Quadratic Equation → Nature of Roots of a Quadratic Equation → Sum and Product of Roots of Quadratic Equation → Quadratic Inequality →

Elimination Method for Simultaneous Equations FAQs

What is the formula for the elimination method for simultaneous equations?

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, x = (c₁b₂ − c₂b₁)/(a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁)/(a₁b₂ − a₂b₁), provided the denominator is non-zero.

When should equations be added in the elimination method?

Add the equations when the coefficients of the variable being eliminated have opposite signs. For example, 3y and −3y cancel when the equations are added.

When should equations be subtracted in the elimination method?

Subtract the equations when the coefficients of the variable being eliminated have the same sign and equal magnitude. For example, 4x − 4x cancels after subtraction.

How do you solve 2x + y = 7 and x − y = 2 by elimination?

Add the equations: 3x = 9, so x = 3. Substitute into x − y = 2: 3 − y = 2, giving y = 1.

What happens if elimination gives 0 = 0?

The two equations are dependent or equivalent, so there are infinitely many solutions. For example, 2x + 4y = 8 and x + 2y = 4 represent the same equation.

What happens if elimination gives 0 = 5?

The equations are inconsistent and have no solution. Their left-hand sides are proportional, but their constant terms are not in the same ratio.

Continue learning Elimination Method for Simultaneous Equations on PrepShots

Continue on PrepShots