BYTETOOLS

System of Equations Solver

Solve systems of 2, 3 or 4 linear equations by Gauss-Jordan elimination. See every step, and get clear answers for unique, none or infinite solutions.

x +
y =
x +
y =
2
x
1
y

Verification

  • Equation 1: LHS = 5, RHS = 5
  • Equation 2: LHS = 1, RHS = 1

Step-by-step elimination

Augmented matrix [A | b]:
[         2          1          5 ]
[         1         -1          1 ]

R1 ÷ 2:
[         1        0.5        2.5 ]
[         1         -1          1 ]

Eliminate column 1 from the other rows:
[         1        0.5        2.5 ]
[         0       -1.5       -1.5 ]

R2 ÷ -1.5:
[         1        0.5        2.5 ]
[        -0          1          1 ]

Eliminate column 2 from the other rows:
[         1          0          2 ]
[        -0          1          1 ]

What is the System of Equations Solver?

The ByteTools System of Equations Solver finds the values of x, y, z (and w) that satisfy a set of linear equations simultaneously.

  • Solves 2×2, 3×3 and 4×4 linear systems
  • Gauss-Jordan elimination with partial pivoting for numerical stability
  • Detects no-solution and infinitely-many-solutions cases from the matrix rank
  • Prints every elimination step as readable text
  • Verification: substitutes the solution back into your equations
  • 100% private — all math runs in your browser

How to use the System of Equations Solver

  1. 1

    Choose the system size: 2×2, 3×3 or 4×4.

  2. 2

    Enter the coefficients of each equation and the constant on the right-hand side.

  3. 3

    Read the solution for each variable, or the no-solution / infinite-solutions verdict.

  4. 4

    Expand the step-by-step elimination to see how the answer was reached.

  5. 5

    Copy the solution with one click.

About the System of Equations Solver

The ByteTools System of Equations Solver finds the values of x, y, z (and w) that satisfy a set of linear equations simultaneously. Choose a 2×2, 3×3 or 4×4 system, type the coefficients and constants, and it solves the system by Gauss-Jordan elimination with partial pivoting — the same method taught in linear algebra courses.

Every elimination step is printed in plain text, so students can follow exactly how the augmented matrix is reduced to row-echelon form. The solver also compares the rank of the coefficient matrix with the rank of the augmented matrix, so it correctly reports when a system has no solution or infinitely many solutions instead of failing silently.

Everything is computed with JavaScript running 100% locally in your browser. No equations are uploaded or stored anywhere, so the tool is instant, private and works offline once the page has loaded.

Frequently asked questions

How do you solve a system of equations with elimination?

Write the system as an augmented matrix, then use row operations — swapping rows, scaling a row, and subtracting multiples of one row from another — until each variable's column has a single 1. The numbers left in the last column are the solution. This tool performs and prints those exact steps.

What does it mean when a system has no solution?

It means the equations contradict each other, like two parallel lines that never cross. Algebraically, elimination produces a row that reads 0 = some non-zero number. The solver detects this by comparing matrix ranks and tells you plainly that the system is inconsistent.

What does infinitely many solutions mean?

It means at least one equation is a combination of the others, so there are fewer independent equations than unknowns. Geometrically the lines or planes overlap. The solver reports this case when the rank of the matrix is smaller than the number of variables.

What is partial pivoting and why does it matter?

Partial pivoting swaps rows so the largest available number is used as the pivot at each step. It prevents dividing by tiny numbers, which would amplify floating-point rounding errors. It makes the answers you get from this solver much more numerically reliable.

Can this solver handle decimals and negative coefficients?

Yes. Coefficients and constants can be any real numbers, including negatives and decimals like -2.5. The solver works in floating-point arithmetic and rounds the displayed solution sensibly while keeping full precision internally.

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