System of Equations Calculator

Equations 3 × 3

The answer updates as you type. Use integers, decimals or fractions like -3/4. An empty box counts as 0.

About the System of Equations Calculator

This calculator solves linear systems of up to ten equations in up to ten unknowns. It keeps every number as an exact fraction, so you get 8/7 instead of 1.142857, and it tells you whether the system has one solution, no solution, or infinitely many.

The number of equations doesn't have to match the number of unknowns. When there are infinitely many solutions, the answer is written in terms of the free variables. Systems with two or three unknowns can also be graphed as lines or planes. Everything runs in your browser.

How to Use the System of Equations Calculator

  1. Set the size. Add or remove equations and variables to match your system.
  2. Type the coefficients. Each row is one equation. Write 2x − y = 5 as 2, -1, 5. If a variable is missing from an equation, leave its box empty.
  3. Read the answer. The Solution tab shows the exact values, the type of system and the rank.
  4. Check the working. The Steps tab shows every row operation on the augmented matrix. The Graph tab draws two-variable systems as lines and three-variable systems as planes.

Methods for Solving Systems of Equations

All of these methods give the same answer. They differ in how much work they take as the system grows.

Substitution

Solve one equation for a variable and substitute it into the others. Quick for 2×2 systems with a coefficient of 1, tedious beyond that.

Elimination

Add multiples of equations together to cancel a variable. This is the hand method taught in algebra classes and the idea behind row reduction.

Gauss–Jordan elimination

Elimination written as row operations on the augmented matrix until it reaches reduced row echelon form. It works for any size and shape, and it is what this calculator uses.

Cramer’s rule

Each unknown is a ratio of two determinants. Handy for 2×2 and 3×3, but it only works when the system is square with a non-zero determinant.

Inverse matrix

Write the system as Ax = b and compute x = A⁻¹b. More work than elimination, and it fails for singular matrices.

Graphing

Plot each equation and see where they meet: lines in 2D, planes in 3D. Good for intuition, imprecise for fractional answers.

One Solution, No Solution, or Infinitely Many?

Compare the rank of the coefficient matrix A with the rank of the augmented matrix [A|b] and the number of unknowns n:

  • One solution: rank A = rank [A|b] = n. The lines or planes meet at exactly one point.
  • Infinitely many: rank A = rank [A|b] < n. Some variables are free, and the solutions form a line, plane or larger set.
  • No solution: rank A < rank [A|b]. Row reduction produces a contradiction like 0 = 1.

Worked Example: A 3×3 System

Solve

2x − y − 3z = −5−3x − y + z = −3−3x − 4y − 2z = −12

Write the coefficients and constants as an augmented matrix, then use row operations to turn the left block into the identity matrix. The right column is then the answer.

  1. StartAugmented matrix [A | b]
    2−1−3−5
    −3−11−3
    −3−4−2−12
  2. Step 1R1 → R1 ÷ 2
    1−12−32−52
    −3−11−3
    −3−4−2−12
  3. Step 2R2 → R2 + 3R1, R3 → R3 + 3R1
    1−12−32−52
    0−52−72−212
    0−112−132−392
  4. Step 3R2 → R2 ÷ (−5/2)
    1−12−32−52
    0175215
    0−112−132−392
  5. Step 4R1 → R1 + (1/2)R2, R3 → R3 + (11/2)R2
    10−45−25
    0175215
    0065185
  6. Step 5R3 → R3 ÷ (6/5)
    10−45−25
    0175215
    0013
  7. Step 6R1 → R1 + (4/5)R3, R2 → R2 − (7/5)R3
    1002
    0100
    0013

Solution: x = 2, y = 0, z = 3. Substitute back into the first equation to check: 2(2) − 1(0) − 3(3) = −5.

Frequently Asked Questions

What is a system of equations?

A system of equations is a set of two or more equations that share the same unknowns. Solving the system means finding values for every unknown that make all of the equations true at the same time.

How many solutions can a linear system have?

Exactly one, none, or infinitely many. Two lines either cross once, never meet (parallel), or lie on top of each other. The same three cases hold for any number of variables, and the calculator tells you which one you have.

Can I enter fractions and decimals?

Yes. Type values such as 3, -2.5, 1/3 or -0.75/2. The solver uses exact rational arithmetic, so answers come back as reduced fractions with no rounding error. Switch to decimals at any time.

What method does the calculator use?

Gauss–Jordan elimination (row reduction to reduced row echelon form). Every row swap, scaling and elimination is shown in the Steps tab, so you can follow or copy the working.

Does the number of equations have to match the number of unknowns?

No. You can solve up to 10 equations in up to 10 unknowns, in any combination. With fewer equations than unknowns the system usually has infinitely many solutions, which the calculator writes in parametric form using the free variables.

Is my data sent anywhere?

No. All calculations run in your browser. Nothing is uploaded and the page keeps working offline once it has loaded.