1
Choose the size
Select a two-variable or three-variable system.
Free Systems of Equations Calculator
A systems of equations calculator finds values that make multiple linear equations true at once. Solve 2 by 2 or 3 by 3 systems and follow every elimination step.
Classroom math support
Turn algebra goals into differentiated lessons
Enter relevant goals and accommodations manually in the lesson planner. Calculator results are not transferred, and lesson drafts should be reviewed before use.
Linear systems
Enter each equation
Use coefficients and constants. Enter 0 when a variable is missing.
Equation 1
x +
y =
Equation 2
x +
y =
System result
x = 2, y = 1
Unique solution: x = 2, y = 1.
Equations entered
2x + y = 5
x - y = 1
Elimination steps
- 1. Write the augmented matrix: [2, 1, 5] [1, -1, 1].
- 2. Divide row 1 by 2: [1, 0.5, 2.5].
- 3. Subtract 1 times row 1 from row 2: [0, -1.5, -1.5].
- 4. Divide row 2 by -1.5: [0, 1, 1].
- 5. Subtract 0.5 times row 2 from row 1: [1, 0, 2].
- 6. Read the solution from the reduced matrix: x = 2, y = 1.
Educational tool: verify important calculations independently. Results use numerical row reduction and may round repeating decimals for display.
How to Solve a System of Equations
2
Enter coefficients
Copy each coefficient and constant into its matching box.
3
Follow elimination
Review every row operation and the final solution type.
Systems of Equations Calculator FAQ
What is a systems of equations calculator?
A systems of equations calculator finds values that satisfy two or more linear equations at the same time. This calculator uses row reduction and shows each elimination step.
How do you solve a system by elimination?
Multiply or divide equations when needed, then add or subtract them to remove one variable. Repeat until each remaining equation isolates a variable, then check the values in the original system.
How can a system have no solution?
A system has no solution when its equations contradict one another. During row reduction, this appears as a row where zero equals a nonzero number.
When does a system have infinitely many solutions?
A system has infinitely many solutions when its equations are dependent and at least one variable remains free after row reduction.
Can this calculator solve three-variable systems?
Yes. Choose three variables, then enter the x, y, and z coefficients and constant for each of three equations.