System of Equations Calculator
Solve an augmented matrix and classify the solution set.
Status
Unique solution
Solution
x1 = -1.4, x2 = -2, x3 = -4.2
Variables
3
Coefficient rank
3
Augmented rank
3
Original system
Augmented matrix written from the input rows.
| Row | x1 | x2 | x3 | b |
|---|---|---|---|---|
| 1 | 2 | -2 | 1 | -3 |
| 2 | 1 | 3 | -2 | 1 |
| 3 | 3 | -1 | -1 | 2 |
Original matrix
Starting matrix.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 2 | -2 | 1 | -3 |
| 2 | 1 | 3 | -2 | 1 |
| 3 | 3 | -1 | -1 | 2 |
Swap row 1 with row 3
Move a stronger pivot into position.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 3 | -1 | -1 | 2 |
| 2 | 1 | 3 | -2 | 1 |
| 3 | 2 | -2 | 1 | -3 |
Eliminate below column 1
Clear the entries below the pivot.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 3 | -1 | -1 | 2 |
| 2 | 0 | 3.3333 | -1.6667 | 0.3333 |
| 3 | 0 | -1.3333 | 1.6667 | -4.3333 |
Eliminate below column 2
Clear the entries below the pivot.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 3 | -1 | -1 | 2 |
| 2 | 0 | 3.3333 | -1.6667 | 0.3333 |
| 3 | 0 | 0 | 1 | -4.2 |
Eliminate below column 3
Clear the entries below the pivot.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 3 | -1 | -1 | 2 |
| 2 | 0 | 3.3333 | -1.6667 | 0.3333 |
| 3 | 0 | 0 | 1 | -4.2 |
Original matrix
Starting matrix.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 2 | -2 | 1 | -3 |
| 2 | 1 | 3 | -2 | 1 |
| 3 | 3 | -1 | -1 | 2 |
Swap row 1 with row 3
Move a stronger pivot into position.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 3 | -1 | -1 | 2 |
| 2 | 1 | 3 | -2 | 1 |
| 3 | 2 | -2 | 1 | -3 |
Scale row 1
Normalize the pivot in column 1.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 1 | -0.3333 | -0.3333 | 0.6667 |
| 2 | 1 | 3 | -2 | 1 |
| 3 | 2 | -2 | 1 | -3 |
Clear column 1
Clear the pivot column above and below the pivot.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 1 | -0.3333 | -0.3333 | 0.6667 |
| 2 | 0 | 3.3333 | -1.6667 | 0.3333 |
| 3 | 0 | -1.3333 | 1.6667 | -4.3333 |
Scale row 2
Normalize the pivot in column 2.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 1 | -0.3333 | -0.3333 | 0.6667 |
| 2 | 0 | 1 | -0.5 | 0.1 |
| 3 | 0 | -1.3333 | 1.6667 | -4.3333 |
Clear column 2
Clear the pivot column above and below the pivot.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 1 | 0 | -0.5 | 0.7 |
| 2 | 0 | 1 | -0.5 | 0.1 |
| 3 | 0 | 0 | 1 | -4.2 |
Clear column 3
Clear the pivot column above and below the pivot.
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 1 | 0 | 0 | -1.4 |
| 2 | 0 | 1 | 0 | -2 |
| 3 | 0 | 0 | 1 | -4.2 |
Formula
A consistent system has the same rank in the coefficient and augmented matrices. A unique solution appears when that rank matches the number of variables.
Use the system calculator to enter one equation per row, including the constant as the last value. The solver reduces the augmented matrix, reports whether the system has one solution, no solution, or infinitely many solutions, and shows the reduction trail.
How to Use
Use the system calculator to enter one equation per row, including the constant as the last value. The solver reduces the augmented matrix, reports whether the system has one solution, no solution, or infinitely many solutions, and shows the reduction trail.
- Open the calculator : Start with System of Equations Calculator.
- Enter values : Fill in the required inputs and any optional settings.
- Review the result : Read the output and use the about page for more detail if needed.
Common Questions
What formula does the System of Equations Calculator use?
A consistent system has the same rank in the coefficient and augmented matrices. A unique solution appears when that rank matches the number of variables.