System of Equations Solver
Solve systems of linear equations with our free System of Equations Solver. Enter coefficients for up to 6 equations with 6 unknowns. Uses Gaussian elimination with back substitution, get step-by-step solutions in plain English, a visual graph, and a detailed matrix breakdown.
What Is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. For example:
2x + 3y = 8
4x − y = 2
Solution: x = 1, y = 2
Linear systems can have:
- One unique solution — the lines intersect at a single point.
- Infinite solutions — the lines are coincident (overlap completely).
- No solution — the lines are parallel (never intersect).
How Does the System of Equations Solver Work?
This solver uses Gaussian elimination with back substitution, one of the most efficient algorithms for solving linear systems.
Step 1: Convert the system to an augmented matrix.
Step 2: Perform row operations to transform the matrix to row echelon form (upper triangular).
Step 3: Use back substitution to find the values of the variables.
Step 4: Classify the system as consistent (unique or infinite solutions) or inconsistent (no solution).
The solver supports systems with up to 6 equations and 6 unknowns. It detects inconsistent systems (where a row becomes 0 = non-zero) and infinite solutions (where a row becomes all zeros).
Why Use This System of Equations Solver?
- Flexible Sizing: Supports 2×2 up to 6×6 systems.
- Visual Graph: For 2×2 systems, see the lines intersect (or not) on a graph.
- Plain English Steps: Each elimination step is explained in simple language.
- Classification: Automatically determines if the system has a unique solution, infinite solutions, or no solution.
- Free & Private: No registration, no data storage.
❓ System of Equations Solver FAQ
What is a system of linear equations?
A system of linear equations is a set of equations where each equation is linear (the variables appear to the first power). The solution is the set of values that makes all equations true simultaneously.
What is Gaussian elimination?
Gaussian elimination is an algorithm for solving systems of linear equations. It transforms the system's augmented matrix into row echelon form using row operations, then uses back substitution to find the solution.
What are the possible solutions to a linear system?
A linear system can have: one unique solution (consistent and independent), infinite solutions (consistent and dependent), or no solution (inconsistent).
What is an augmented matrix?
An augmented matrix combines the coefficient matrix and the constant terms into a single matrix. For example, the system 2x + 3y = 8 and 4x − y = 2 becomes [2 3 | 8; 4 -1 | 2].
What does "inconsistent system" mean?
An inconsistent system has no solution. This happens when the equations contradict each other, typically when elimination produces a row like 0 = 5.
What does "infinite solutions" mean?
A system has infinite solutions when the equations are dependent (one equation is a multiple of another). This results in a free variable, and the solution can be expressed in terms of that variable.
What is back substitution?
Back substitution is the process of solving for variables from the bottom row upward after the matrix has been reduced to row echelon form. The last row gives the value of the last variable, which is then substituted into the row above.
What is the maximum system size this calculator supports?
This calculator supports up to 6 equations with 6 unknowns (6×6). You can select sizes from 2×2 to 6×6 using the size buttons.
How do I enter coefficients?
Enter the coefficient for each variable in each equation, followed by the constant term on the right side. For example, for 2x + 3y - z = 10, enter 2, 3, -1, 10.
What does the graph show?
For 2×2 systems, the graph shows the two lines representing each equation. The intersection point (if any) is the solution. For larger systems, the graph shows a bar chart of the solution values.
Is this calculator free to use?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.