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Linear Combination Calculator

Solve two linear equations in two variables with elimination.

Linear Combination Calculator Inputs

Results computed instantly — your data never leaves your device.

Live Linear Combination Calculator Results

Real-Time

System result

x = -3.000000, y = -1.000000

The determinant guard distinguishes unique, inconsistent, and dependent systems.

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How to Use the Linear Combination Calculator

  1. 1

    Enter coefficients for the first equation.

  2. 2

    Enter coefficients for the second equation.

  3. 3

    Review x and y when the system has one solution.

  4. 4

    Read the guard message for dependent or inconsistent systems.

Formula & Mathematical Basis

For a₁x+b₁y=c₁ and a₂x+b₂y=c₂, solve using determinant D=a₁b₂−a₂b₁.

Variable Key

a₁,b₁,c₁

Coefficients of the first equation

a₂,b₂,c₂

Coefficients of the second equation

📝 A zero determinant is classified as infinitely many solutions when equations are consistent, otherwise no solution.

Step-by-Step Examples

1

Two-equation system

Scenario: Use x−4y=1 and −2x+4y=2.

  1. 1.Enter all six coefficients.
  2. 2.Calculate the determinant.
  3. 3.Use elimination or the equivalent determinant formulas.
  4. 4.Check x and y in both equations.
✅ The result card reports the unique solution or the appropriate system classification.

Practical Use Cases

  • Practice elimination of two linear equations.
  • Check algebra homework.
  • Identify dependent and inconsistent systems.

Common Pitfalls

  • Swapping a coefficient with the constant.
  • Dividing by a zero determinant.
  • Failing to verify the solution in both equations.

Frequently Asked Questions

What does zero determinant mean?

It means the system is not uniquely solvable; it may have no solution or infinitely many.

Are intermediate elimination steps shown?

The current route reports the classification or solution; it does not reproduce another publisher’s step-by-step layout.

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