Elimination Method

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

Intermediate25 minLesson

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.

Adding these equations:

The terms ( and ) 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:

Identify opposite coefficients

The coefficients are and - they're opposites! → Ready to add

Add the equations

→

Solve for

→

Substitute back to find

→

Verify the solution

Equation 1: ✓ Equation 2: ✓ → Both equations satisfied

Answer: The solution is

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 by , you get , not .

Adding when you should subtract (or vice versa)

Why it's wrong: For elimination to work, the coefficients must be true opposites (like and ). 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.

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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 = VIP tickets and = regular tickets. System: (total tickets) (total revenue)

How many VIP tickets were sold?

Step 1: Write the mathematical expression

Multiply the first equation by and add to eliminate :

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 = mL of 20% solution and = mL of 50% solution. System: (total volume) (acid amount: mL)

How many mL of the 50% solution are needed?

Step 1: Write the mathematical expression

Multiply the first equation by 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 ).

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 and
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