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Solve The Following System of Linear Equations Calculator

Reviewed by Calculator Editorial Team

Solving systems of linear equations is a fundamental skill in algebra and mathematics. This calculator helps you solve systems with 2, 3, or more variables using methods like substitution, elimination, and matrix operations.

How to Use This Calculator

To solve a system of linear equations:

  1. Enter the number of variables in your system (2, 3, or more).
  2. Input each equation in the format "ax + by + cz = d".
  3. Click "Calculate" to solve the system.
  4. Review the solution and interpretation.

The calculator will determine if the system has a unique solution, infinitely many solutions, or no solution.

Formula Used

For a system of linear equations in matrix form Ax = b:

If the determinant of A is non-zero, the unique solution is x = A⁻¹b.

If the determinant is zero, the system may have infinitely many solutions or no solution.

The calculator uses Gaussian elimination for systems with more than 3 variables.

Worked Example

Solve the system:

2x + 3y = 8

4x - y = 2

Using substitution:

  1. From the second equation: y = 4x - 2
  2. Substitute into the first equation: 2x + 3(4x - 2) = 8
  3. Solve for x: 2x + 12x - 6 = 8 → 14x = 14 → x = 1
  4. Find y: y = 4(1) - 2 = 2

The solution is x = 1, y = 2.

Frequently Asked Questions

How many variables can this calculator solve?

This calculator can solve systems with 2, 3, or more variables using appropriate methods.

What if my system has no solution?

The calculator will indicate if the system is inconsistent and has no solution.

Can I solve nonlinear systems with this calculator?

No, this calculator is designed for linear systems only.