Elimination Method Calculator

Solve systems of linear equations using the elimination method with step-by-step solutions

System of Linear Equations

Current System:

3x - 4y = 6
-x + 4y = 2

First Equation: a₁x + b₁y = c₁

Second Equation: a₂x + b₂y = c₂

Solution

x = 4.000000
X Value
y = 1.500000
Y Value

Solution Type

Unique Solution

Solution: x = 4.000000, y = 1.500000

Step-by-Step Solution

Original system:
3x -4y = 6
-1x +4y = 2
Eliminating x by using LCM(3, -1) = 3:
Multiply equation 2 by 3: -3x +12y = 6
Adding equations: 8y = 12
Solving for y: y = 12/8 = 1.500000
Substituting y = 1.500000 into equation 1:
3x -4(1.500000) = 6
x = (6 - -6)/3 = 4.000000
Verification:
Equation 1: 3(4.000000) -4(1.500000) = 6.000000 ✓
Equation 2: -1(4.000000) +4(1.500000) = 2.000000 ✓

Example Problems

Example 1: Simple Elimination

System: 3x - 4y = 6, -x + 4y = 2

Method: Add equations directly (y coefficients are opposites)

Result: 2x = 8, so x = 4, y = 1.5

Example 2: Using Multipliers

System: 2x + 3y = 5, 2x + 7y = -3

Method: Multiply first equation by -1, then add

Result: 4y = -8, so y = -2, x = 5.5

Example 3: No Solution

System: -4x + 8y = 5, 3x - 6y = -1

Method: After elimination, get 0 = 11 (false)

Result: No solution (inconsistent system)

Elimination Method Steps

1

Arrange Equations

Ensure variables are aligned

2

Create Opposites

Multiply equations to get opposite coefficients

3

Add Equations

Eliminate one variable

4

Solve & Substitute

Find remaining variable, then substitute back

Solution Types

Unique Solution

One specific (x, y) pair

Infinite Solutions

Dependent equations (same line)

No Solution

Inconsistent system (parallel lines)

Understanding the Elimination Method

What is the Elimination Method?

The elimination method is a systematic approach to solving systems of linear equations by eliminating one variable at a time. By strategically adding equations together, we can eliminate a variable and reduce the system to a simpler form that's easier to solve.

When to Use Elimination

  • When coefficients are already opposites or can be made opposites easily
  • For systems with integer coefficients
  • When substitution would result in fractions or complex expressions

Key Strategies

Creating Opposites

Use LCM to find multipliers that create opposite coefficients

Choose Wisely

Eliminate the variable that requires simpler multipliers

Verify Solution

Always check your answer in both original equations

Special Cases

1

Unique Solution

Lines intersect at exactly one point

Infinite Solutions

Same line (dependent equations)

No Solution

Parallel lines (inconsistent system)