Elimination Method

Learn to solve systems of equations by eliminating one variable through addition or subtraction.

Intermediate25 minLesson
Printable worksheet

Definition

The elimination method (also called the addition method) solves systems of equations by adding or subtracting the equations to eliminate one variable.
Key Idea: When we add two equations, we can make one variable disappear if its coefficients are opposites.
{2x+3y=122x−3y=0\begin{cases} 2x + 3y = 12 \\ 2x - 3y = 0 \end{cases}
Adding these equations:
4x+0y=124x + 0y = 12
x=3x = 3
The yy terms (+3y+3y and −3y-3y) are opposites, so they cancel out when we add!

Try it now

Which system is ready for elimination by adding (no multiplication needed)?

Worked Examples

Solve the system:
{x+2y=7x−2y=1\begin{cases} x + 2y = 7 \\ x - 2y = 1 \end{cases}
1

Identify opposite coefficients

The yy coefficients are +2+2 and −2-2 - they're opposites! → Ready to add
2

Add the equations

(x+2y)+(x−2y)=7+1(x + 2y) + (x - 2y) = 7 + 1 2x+0y=82x + 0y = 8 → 2x=82x = 8
3

Solve for xx

2x=82x = 8 x=4x = 4
4

Substitute back to find yy

4+2y=74 + 2y = 7 2y=32y = 3 y=1.5y = 1.5
5

Verify the solution

Equation 1: 4+2(1.5)=4+3=74 + 2(1.5) = 4 + 3 = 7 ✓ Equation 2: 4−2(1.5)=4−3=14 - 2(1.5) = 4 - 3 = 1 ✓ → Both equations satisfied

Common Mistakes

Forgetting to multiply the entire equation

Why it's wrong: When multiplying to create opposite coefficients, every term must be multiplied, including the constant on the right side.

Correct: If multiplying x+2y=5x + 2y = 5 by 33, you get 3x+6y=153x + 6y = 15, not 3x+6y=53x + 6y = 5.

Adding when you should subtract (or vice versa)

Why it's wrong: For elimination to work, the coefficients must be true opposites (like +3+3 and −3-3). If they're the same, you need to subtract instead.

Correct: Same signs: subtract the equations. Opposite signs: add the equations.

Not verifying the solution in both equations

Why it's wrong: A solution must satisfy both original equations. Checking only one equation can miss errors.

Correct: Always substitute your answer into BOTH original equations to verify.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Solving Systems of Equations by Elimination | Step-by-Step Examples

by Understand The Math

Watch on YouTube

Interactive Visual

Function graph

-2024680510xy(4, 2)

Tap or hover over the graph to read off values.

3x + 2y = 16x + y = 6
  • Intersection point: (4, 2)

Equation Solver

Solve for x,yx, y:

(1)x+2y=7(2)x−2y=1\begin{aligned} (1)\quad x + 2y &= 7 \\ (2)\quad x - 2y &= 1 \end{aligned}
1
(1)x+2y=7(2)x−2y=1\begin{aligned} (1)\quad x + 2y &= 7 \\ (2)\quad x - 2y &= 1 \end{aligned}
Subtract equation (2) from equation (1)The xx terms cancel out

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = (16 - 3x)/2

m=-1.5, b=8

y = 6 - x

m=-1, b=6

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

16 problems
Problem 1 of 16
Easy
Which system is ready for elimination by adding (no multiplication needed)?

Why It Matters

The elimination method is powerful because:
  • Efficient for certain systems: When coefficients are already opposites or equal, elimination is faster than substitution
  • No fractions needed: Often avoids messy fractions that substitution creates
  • Foundation for advanced math: Matrix operations in linear algebra use the same concept
  • Real applications: Engineers use systems of equations to analyze circuits, balance chemical equations, and optimize resources
Choosing between substitution and elimination is like choosing the right tool for a job - both work, but one may be easier for a particular problem.

Real World Applications

Concert Ticket Sales

Event managers use systems of equations to analyze ticket sales when different types have different prices.

Example:

A concert sold 500 tickets total. VIP tickets cost 80 euros and regular tickets cost 40 euros. Total revenue was 28,000 euros. How many of each type were sold?

1Try It Yourself

Let vv = VIP tickets and rr = regular tickets. System: v+r=500v + r = 500 (total tickets) 80v+40r=2800080v + 40r = 28000 (total revenue)

How many VIP tickets were sold?

Step 1: Write the mathematical expression

Multiply the first equation by −40-40 and add to eliminate rr:

Chemistry: Balancing Mixtures

Chemists use elimination to determine how much of each solution to mix to achieve a desired concentration.

Example:

A chemist needs 100 mL of a 30% acid solution. She has 20% and 50% acid solutions available. How much of each should she mix?

2Try It Yourself

Let xx = mL of 20% solution and yy = mL of 50% solution. System: x+y=100x + y = 100 (total volume) 0.20x+0.50y=300.20x + 0.50y = 30 (acid amount: 100×0.30=30100 \times 0.30 = 30 mL)

How many mL of the 50% solution are needed?

Step 1: Write the mathematical expression

Multiply the first equation by −0.20-0.20 and add:

Key Takeaways

  • 1The elimination method eliminates one variable by adding or subtracting equations
  • 2If coefficients are already opposites, add the equations directly
  • 3If coefficients are the same, subtract the equations
  • 4If neither, multiply one or both equations to create opposite coefficients
  • 5Always verify your solution by substituting into both original equations

Frequently Asked Questions

Use elimination when: (1) the coefficients are already opposites or equal, (2) neither equation is already solved for a variable, or (3) substitution would create messy fractions. Use substitution when one equation is already solved for a variable (like y=3x+2y = 3x + 2).
Use elimination when: (1) the coefficients are already opposites or equal, (2) neither equation is already solved for a variable, or (3) substitution would create messy fractions. Use substitution when one equation is already solved for a variable (like y=3x+2y = 3x + 2).
If you get a true statement like 0=00 = 0, the system has infinitely many solutions (the lines are the same). If you get a false statement like 0=50 = 5, the system has no solution (the lines are parallel).
No! You'll get the same answer either way. Choose the variable that requires the simplest multiplication to create opposite coefficients.

Glossary

Elimination method
A technique for solving systems of equations by adding or subtracting equations to eliminate one variable
Opposite coefficients
Coefficients that sum to zero, like +3+3 and −3-3
LCM (Least Common Multiple)
The smallest number that both coefficients divide into evenly; used to find the multipliers needed to create opposite coefficients
Verify
To check that a solution satisfies both original equations

Formula Card

Elimination Strategy

Opposites: +a and −a→ADD\text{Opposites: } +a \text{ and } -a \rightarrow \text{ADD}
Same: +a and +a→SUBTRACT\text{Same: } +a \text{ and } +a \rightarrow \text{SUBTRACT}

Decide whether to add or subtract based on coefficients

Creating Opposites

Multiply equations to get ±LCM of coefficients\text{Multiply equations to get } \pm\text{LCM of coefficients}

Use the LCM to find multipliers

Today's challengeOne problem, three tries

More in This Topic

Introduction to Systems of Equations

Intermediate25 min

Systems of Equations (Substitution)

Advanced25 min

Solving Systems by Graphing

Intermediate25 min

Substitution Method

Intermediate25 min