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Algebra / Systems of Equations

Elimination Method

The elimination method solves a system by adding or subtracting the equations so that one variable cancels out. When both equations are in standard form, it is usually the fastest route to the answer and it never forces you into fractions early.

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Chapter 7 · Lesson 3 of 11Systems of Equations
  1. 1.Systems of Linear Equations
  2. 2.Substitution Method
  3. 3.Elimination Method
  4. 4.Graphing Systems of Equations
  5. 5.Systems of Equations With Three Variables
  6. 6.Augmented Matrices
  7. 7.Cramer's Rule
  8. 8.Inverse of a Matrix
  9. 9.Nonlinear Systems of Equations
  10. 10.Systems of Linear Inequalities
  11. 11.Systems of Equations Word Problems
See all algebra chapters →

Substitution works on any system, but it can get messy. Try solving and by isolating a variable and you’ll be juggling fractions right away. The elimination method takes a different route. Instead of rearranging one equation, it combines the two equations so that one variable cancels completely. It’s sometimes called the addition method or “solving by combining the equations”, and for systems in standard form it’s usually the fastest method there is.

Elimination method: if b1 = −b2, adding the equations gives (a1 + a2)x = c1 + c2
Elimination method

What Is the Elimination Method?

Line up the two equations of a system of linear equations one above the other, with like terms in the same columns:

If the -coefficients are opposites, so that , adding the equations column by column wipes out :

Why is adding equations allowed? Each equation says its left side and right side are the same number. Adding the same number to both sides of the first equation keeps it true, and the second equation hands you that number in two forms. So the sum of two true equations is still true.

The equations 3x + 2y = 16 and 5x − 2y = 16 added column by column: the +2y and −2y cancel, leaving 8x = 32, so x = 4 and then y = 2
Opposite coefficients cancel when the equations are added

Steps of the Elimination Method

  1. Line up both equations in standard form, , with like terms in columns.
  2. Make opposites. If no variable already has opposite (or equal) coefficients, multiply one or both equations by a constant until one does.
  3. Add or subtract the equations to eliminate that variable. Add when the coefficients are opposites; subtract when they’re equal.
  4. Solve the one-variable equation that’s left.
  5. Back-substitute into either original equation to find the other variable, then check in both.

Solving Systems by Elimination: Examples

Example 1: Coefficients Already Opposite

Solve and .

Step 1 — the -terms are and , so add the equations:

Step 2 — back-substitute into the first equation:

Step 3 — check in the second equation:

Answer:

Example 2: Equal Coefficients, So Subtract

Solve and .

Step 1 — both -terms are , so subtract the second equation from the first:

Step 2 — back-substitute:

Answer: . Subtract every term, including the constants: , not .

Example 3: Multiply One Equation

Solve and .

Step 1 — no coefficients match yet. Multiply the second equation by so that becomes :

Step 2 — add it to the first equation:

Step 3 — back-substitute:

Answer:

Example 4: Multiply Both Equations

Solve and , the system from the introduction.

Step 1 — target . The coefficients and have least common multiple . Multiply the first equation by and the second by :

Step 2 — add:

Step 3 — back-substitute into :

Step 4 — check in the second original equation:

Answer:

3x + 4y = 10 multiplied by 3 becomes 9x + 12y = 30, and 5x − 6y = 4 multiplied by 2 becomes 10x − 12y = 8; adding them gives 19x = 38, so x = 2 and y = 1
Scale both equations until the y-terms are +12y and −12y, then add

Multiply every term, constants included. is ; forgetting to multiply the is the classic slip.

Elimination With No Solution or Infinitely Many Solutions

If both variables cancel at once, look at what’s left, just as with substitution.

No solution. Solve and . Multiply the first equation by to get , then subtract the second equation:

That’s false, so the system has no solution. The lines are parallel.

Infinitely many. Solve and . Multiply the first equation by to get , then subtract:

That’s always true, so the two equations are the same line and there are infinitely many solutions.

Substitution vs Elimination

Both methods always reach the same answer. The question is which gets you there with less work.

SubstitutionElimination
Core movesolve one equation for a variable, plug it into the otheradd or subtract the equations to cancel a variable
Best whena variable is alone, or has coefficient or both equations in standard form, larger coefficients
Fractionscan appear earlyusually avoided until the end
Works for nonlinear systemsyesonly in special cases
Extends to three variablespossible but slowthe standard method
A four-step decision card: if a variable is already alone, use substitution; if a variable has coefficient 1 or −1, use substitution or elimination; if two matching coefficients are opposites or equal, add or subtract; otherwise multiply, then eliminate
A quick way to choose between the substitution and elimination methods

If you’re not sure, elimination is the safer default for systems in standard form. The substitution method is the better tool when one equation already reads .

Worked Example: A Word Problem

At a coffee shop, 3 lattes and 2 muffins cost , while 2 lattes and 4 muffins cost . How much is one latte, and how much is one muffin?

Step 1 — define the variables: let be the price of a latte and the price of a muffin.

Step 2 — write the system:

Step 3 — target . Multiply the first equation by so both have :

Step 4 — the -coefficients are now equal, so subtract the second equation:

Step 5 — back-substitute into the first equation:

Step 6 — check in the second equation:

A latte costs and a muffin costs .

The lines 3L + 2M = 19.50 and 2L + 4M = 19.00 on axes of latte price and muffin price, crossing at (5, 2.25)
Both price equations are true only at a $5 latte and a $2.25 muffin

Common Mistakes to Avoid

  • Multiplying only part of an equation. When you scale an equation, every term changes, including the constant on the right.
  • Adding when you should subtract. Equal coefficients ( and ) need subtraction; opposite ones ( and ) need addition.
  • Sign errors in subtraction. Subtracting means subtracting , which adds . Put the second equation in brackets: .
  • Not lining up the columns. Rewrite as before eliminating, so sits under .
  • Stopping at one variable. Elimination gives you one value. Back-substitute for the other, then check both equations.

Practice Problems

Work each one before opening the answer.

Problem 1. Solve by elimination: and .

Show answer

Step 1 — the -terms are opposites, so add:

Step 2 — back-substitute:

Answer:

Problem 2. Solve by elimination: and .

Show answer

Step 1 — the -terms are equal, so subtract the second equation from the first:

Step 2 — back-substitute:

Answer:

Problem 3. Solve by elimination: and .

Show answer

Step 1 — multiply the second equation by :

Step 2 — subtract it from the first equation:

Step 3 — back-substitute into :

Answer:

Problem 4. Solve by elimination: and .

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Step 1 — target . The LCM of and is , so multiply the first equation by and the second by :

Step 2 — subtract the first from the second:

Step 3 — back-substitute:

Answer:

Problem 5. Solve by elimination: and .

Show answer

Step 1 — multiply the first equation by :

Step 2 — subtract the second equation:

Step 3 — a false statement means the lines are parallel.

Answer: no solution

Problem 6. A jar holds dimes and quarters worth in total. How many of each coin are there?

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Step 1 — write the system, with dimes and quarters, working in cents:

Step 2 — multiply the first equation by and subtract it from the second:

Step 3 — back-substitute:

Answer: dimes and quarters

Elimination really shows its strength once systems grow beyond two variables. With three equations and three unknowns, you eliminate one variable twice to get down to a two-variable system, and then finish it with exactly the steps on this page.

Frequently Asked Questions

What is the elimination method?+

It solves a system of equations by combining the equations so that one variable disappears. If the -coefficients are opposites, such as and , adding the equations removes and leaves an equation in alone. If they aren't opposites yet, you multiply one or both equations first.

Why is it allowed to add two equations together?+

Each equation says two quantities are equal. Adding equal amounts to both sides of an equation keeps it balanced, and the second equation gives you exactly that: its left side equals its right side. So the new equation is true whenever both originals are.

When do you add and when do you subtract?+

Add when the coefficients of a variable are opposites, like and . Subtract when they are equal, like and . Either way, that variable disappears.

How do you choose what to multiply by?+

Pick a variable and find the least common multiple of its two coefficients. Multiply each equation so that the variable's coefficient becomes that LCM, with opposite signs. For and , the LCM is , so multiply the first equation by and the second by .

Is elimination better than substitution?+

Neither is always better. Elimination is usually faster when both equations are in standard form with no coefficient of . Substitution is faster when a variable is already alone. Both give the same solution.

Related lessons

Systems of Linear EquationsSubstitution MethodStandard Form of a Linear Equation: Ax + By = CLinear Equations

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