Solving Systems of Linear Equations

Systems of linear equations are fundamental in algebra and appear in various applications from engineering to economics. A system has a unique solution if the number of equations matches the number of variables and the equations are independent.

Using This Calculator

Select 2x2 or 3x3 system size
Enter coefficients for each variable
Click Solve to compute solutions
Results show values and solution method

Examples

2x2 Example: 2x + y = 5, x - y = 1 (Solution: x=2, y=1)

3x3 Example: x+y+z=6, 2y+5z=-4, 2x+5y-z=27 (Solution: x=5, y=3, z=-2)

Methods Used

2x2 Systems: Matrix inversion method using determinants

3x3 Systems: Gaussian elimination with partial pivoting

Reference Table: Matrix Properties

System Size	Required Condition	Method
2x2	ad - bc ≠ 0	Determinant inversion
3x3	Non-zero determinant	Gaussian elimination

Tips & Warnings

For singular matrices (determinant = 0), no unique solution exists
Results are rounded to 4 decimal places for clarity
Large coefficients may cause floating-point precision issues

System of Equations Calculator

Solution

Solving Systems of Linear Equations

Using This Calculator

Examples

Methods Used

Reference Table: Matrix Properties

Tips & Warnings

Gaussian Elimination Steps

Frequently Asked Questions

What if my matrix is singular?

What precision are results shown in?

Can I solve larger systems?

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System of Equations Calculator - 2x2 and 3x3 Solvers

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