Elimination Method: Definition, Steps & Examples

Definitionnoun phrase

What is the elimination method in math?

The elimination method solves a system of equations by adding or subtracting them so one variable cancels. Scaling one equation first is often needed to make the coefficients match. After one variable is eliminated, solve for the other variable, then substitute back to find the first.

In Spanish: método de eliminación
Also called: linear combination, addition method, elimination
Grade level: Algebra 1

Grade 8Algebra 1CCSS 8.EE.C.8.bCCSS HSA-REI.C.5GEDSATACT

Elimination Method in a picture

Opposite coefficients add to zero, so adding the equations removes y and leaves an equation in x alone. The answer is the point where both lines cross.

Formal definition and notation

For a system of equations in two variables, elimination replaces one equation with the sum of that equation and a multiple of the other. This new system has exactly the same solutions. The multiple is chosen so the coefficients of one variable become opposites, like +2y and −2y, and add to 0. If both variables cancel, 0 = 0 means infinitely many solutions and a false statement such as 0 = 7 means no solution.

Symbol	Read it as	Meaning
Ax + By = C	A x plus B y equals C	standard form; line up the x-terms, y-terms and constants before eliminating
(x, y)	the ordered pair x, y	the solution: the values that make both equations true

Key facts

Elimination works best when the equations are in standard form, Ax + By = C.
Multiplying a whole equation by a nonzero number does not change its solutions.
The solution (x, y) is where the two lines cross on a graph.
0 = 0 after eliminating: the equations describe the same line, so there are infinitely many solutions.
A false result such as 0 = 7: the lines are parallel, so there is no solution.

Elimination steps

make one pair of coefficients opposites, then add the equations

1: Line up like terms: x-terms, y-terms, equals sign, constants.
2: Multiply one or both equations (every term, both sides) so the x- or y-coefficients are opposites.
3: Add the equations; one variable drops out. Solve for the other.
4: Substitute back into an original equation to find the second variable, then check both equations.

If the coefficients are already equal (not opposite), you can subtract the equations instead of adding.

Elimination Method examples, step by step

Example 1: Opposites are ready

Warm-up

Solve x + y = 10 and x − y = 4.

The y-terms, +y and −y, are already opposites. Add the equations: 2x = 14.
Divide by 2: x = 7.
Substitute into x + y = 10: 7 + y = 10, so y = 3.

Answer: x = 7 and y = 3, the point (7, 3).

Example 2: Multiply first

Core

Solve 2x + 3y = 12 and 4x − y = 10.

Make the y-terms opposites: multiply the second equation by 3, every term: 12x − 3y = 30.
Add it to the first equation: 14x = 42, so x = 3.
Substitute into 4x − y = 10: 12 − y = 10, so y = 2.
Check in the first equation: 2(3) + 3(2) = 12.

Answer: x = 3 and y = 2, the point (3, 2).

Example 3: A word problem

Test-style

3 notebooks and 2 pens cost 13 dollars. 5 notebooks and 2 pens cost 19 dollars. Find the price of a notebook and of a pen.

Let n be the notebook price and p the pen price: 3n + 2p = 13 and 5n + 2p = 19.
The p-terms are equal, so subtract the first equation from the second: 2n = 6, so n = 3.
Substitute: 3(3) + 2p = 13, so 2p = 4 and p = 2.
Check: 5(3) + 2(2) = 19.

Answer: A notebook costs 3 dollars and a pen costs 2 dollars.

Common mistakes

✗ MistakeAdding 2x + 3y = 7 and x + 3y = 5 to cancel y.

✓ InsteadAdding gives 3x + 6y = 12; nothing cancels. Subtract instead: x = 2.

Make sure the eliminated terms have the same coefficient with opposite signs, or make them match first. Equal coefficients cancel by subtracting.

✗ MistakeMultiplying only the left side: 3(4x − y) = 10 becomes 12x − 3y = 10.

✓ InsteadMultiply every term on both sides: 12x − 3y = 30.

An equation stays balanced only if both sides are multiplied by the same number.

✗ MistakeSubtracting x − 2y = 1 from 3x + 4y = 11 as 2x + 2y = 10.

✓ InsteadSubtract every term: 3x − x = 2x, 4y − (−2y) = 6y, 11 − 1 = 10, so 2x + 6y = 10.

Subtracting a negative term means adding it. Many students subtract only the first term correctly.

✗ MistakeStopping at x = 5.

✓ InsteadSubstitute back to find y, and write the solution as an ordered pair, (5, 2).

A system’s solution needs a value for every variable.

Not to be confused with

Elimination Methodvs.Substitution method

Substitution solves one equation for a variable and plugs it into the other, which is easiest when a variable has coefficient 1, like y = 2x − 3. Elimination adds or subtracts whole equations and suits standard form.

Elimination Methodvs.System of equations

A system is the set of equations; elimination is one way to solve it, along with substitution and graphing.

Quick check: 3 questions

Question 1 of 3

Adding 5x + 2y = 1 and 3x − 2y = 15 gives which equation?

Show answer and explanation

Answer: C. 5x + 3x = 8x, 2y + (−2y) = 0, and 1 + 15 = 16, so 8x = 16 and x = 2. Choice A adds 2y and 2y instead of 2y and −2y, choice B subtracts, and choice D subtracts the constants.

Question 2 of 3

To eliminate x by adding, what should you multiply 2x + 5y = 1 by, if the other equation is 6x − y = 11?

Show answer and explanation

Answer: D. Multiplying by −3 gives −6x, the opposite of 6x, so adding cancels x. Multiplying by 3 gives 6x, which would need subtracting, and 6 or −2 do not match 6x.

Question 3 of 3

While solving a system by elimination, both variables cancel and you get 0 = 7. What does that mean?

Show answer and explanation

Answer: B. A false statement means no pair (x, y) makes both equations true: the lines are parallel. 0 = 0 would mean infinitely many solutions, and 0 = 7 is not a point or a value of x.

Elimination Method: frequently asked questions

When should you use elimination instead of substitution?

Use elimination when both equations are in standard form, Ax + By = C, and no variable has a coefficient of 1 or −1, as in 4x + 3y = 10 and 6x − 3y = 5. Use substitution when one equation is already solved for a variable, such as y = 2x + 1. Both methods always give the same answer.

Do you add or subtract the equations?

Add when the coefficients of one variable are opposites, such as +4y and −4y. Subtract when they are the same, such as +4y and +4y. Many teachers suggest always making them opposites and adding, because subtracting a whole equation is where sign mistakes happen.

Why does the elimination method work?

If (x, y) makes both equations true, then it also makes their sum true, because you are adding equal amounts to both sides of an equation. So the new equation has the same solution. Choosing multiples that make one variable’s coefficients add to 0 leaves an equation with one variable, which you can solve directly.

Practice and tools

Effortless Math practicePractice on Effortless Math System of Equations CalculatorSolve a system step by step and check your elimination Cramer's Rule CalculatorAnother way to solve the same systems

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.C.8.b, CCSS HSA-REI.C.5

Cite this entry

Effortless Math Team. “Elimination Method.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/elimination-method/

Flashcard

TermElimination Methodadd equations to cancel a variable

DefinitionSolve a system by adding or subtracting the equations so one variable cancels. x + y = 10 plus x − y = 4 gives 2x = 14.

Study the full deck

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Elimination Method

What is the elimination method in math?

Elimination Method in a picture

Formal definition and notation

Key facts

Elimination steps

Elimination Method examples, step by step

Example 1: Opposites are ready

Example 2: Multiply first

Example 3: A word problem

Common mistakes

Not to be confused with

Quick check: 3 questions

Elimination Method: frequently asked questions

When should you use elimination instead of substitution?

Do you add or subtract the equations?

Why does the elimination method work?

Related terms

Practice and tools

Sources

Flashcard

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