Maths / Linear algebra
System of Equations Solver: 2–4 Unknowns with Steps
Solve linear systems with two, three or four unknowns using exact fractions. See elimination steps and distinguish a unique answer, no solution and infinitely many solutions.
System of Equations Solver: 2–4 Unknowns with Steps: Simultaneous equations ask for values that satisfy every equation at once. This solver starts with a three-variable example and shows x₁ = 2, x₂ = 3 and x₃ = −1. Replace the augmented matrix with your own coefficients and right-hand sides. For 2x + y = 7, the row is 2 1 7; keep the same variable order in every row. Missing variables need a zero coefficient. The last entry is always the right-hand side, not another unknown. Elimination scales rows and subtracts multiples of one row from another until each leading variable is isolated. Fractions and entered decimals are kept exactly throughout those operations, so a result such as 2/3 stays 2/3. Use the [matrix calculator](/matrix-calculator) when you also need products, determinants or inverses, and the [scientific calculator](/scientific-calculator) to combine the answer with other functions. Runs 100% locally in your browser with zero server file uploads.
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- School & study tools
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- In your browser
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- Free · no sign-up
- Availability
- Ready to use
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More functions, constants and conversions
Use semicolons between arguments: ncr(52;5), dms(30;15;30), randint(1;6). Rnd rounds to the selected significant digits. E is Euler's number until you store a value in E; C is the speed of light until you store C.
- GCD
- 6
- LCM
- 144
- Integer quotient and remainder
- 2; 12
- 48 = 2 × 18 + 12
- 18 = 1 × 12 + 6
- 12 = 2 × 6 + 0
Constants: c, h, hbar (ħ), qe (elementary charge), me, mp, na, kb, gasr, grav (G), g (standard gravity), eps0, mu0, amu and sigma. Measured constants carry uncertainty; the displayed values are not exact. NIST / CODATA constants
Type or tap. Multiplication can be implied, as in 2π or 3(4+1); ^ is a power, ! a factorial, and % divides by 100. Enter calculates; Escape clears.
- x1
- 2 ≈ 2
- x2
- 3 ≈ 3
- x3
- -1 ≈ -1
Elimination steps
R1 ÷ 2
| 11 | 1/20.5 | -1/2-0.5 | 44 |
| -3-3 | -1-1 | 22 | -11-11 |
| -2-2 | 11 | 22 | -3-3 |
R2 − -3 × R1
| 11 | 1/20.5 | -1/2-0.5 | 44 |
| 00 | 1/20.5 | 1/20.5 | 11 |
| -2-2 | 11 | 22 | -3-3 |
R3 − -2 × R1
| 11 | 1/20.5 | -1/2-0.5 | 44 |
| 00 | 1/20.5 | 1/20.5 | 11 |
| 00 | 22 | 11 | 55 |
R2 ÷ 1/2
| 11 | 1/20.5 | -1/2-0.5 | 44 |
| 00 | 11 | 11 | 22 |
| 00 | 22 | 11 | 55 |
R1 − 1/2 × R2
| 11 | 00 | -1-1 | 33 |
| 00 | 11 | 11 | 22 |
| 00 | 22 | 11 | 55 |
R3 − 2 × R2
| 11 | 00 | -1-1 | 33 |
| 00 | 11 | 11 | 22 |
| 00 | 00 | -1-1 | 11 |
R3 ÷ -1
| 11 | 00 | -1-1 | 33 |
| 00 | 11 | 11 | 22 |
| 00 | 00 | 11 | -1-1 |
R1 − -1 × R3
| 11 | 00 | 00 | 22 |
| 00 | 11 | 11 | 22 |
| 00 | 00 | 11 | -1-1 |
R2 − 1 × R3
| 11 | 00 | 00 | 22 |
| 00 | 11 | 00 | 33 |
| 00 | 00 | 11 | -1-1 |
Systems use exact rational elimination. Cubic and quartic roots are numerical approximations checked against the polynomial; repeated or closely spaced roots are less accurate.
Linear systems
OpenStax College Algebra, Systems of Linear Equations: Three Variables (https://openstax.org/books/college-algebra-2e/pages/7-2-systems-of-linear-equations-three-variables), introduces simultaneous equations and dependent systems. Its worked examples use elimination.
Row operations
OpenStax College Algebra, Solving Systems with Gaussian Elimination (https://openstax.org/books/college-algebra-2e/pages/7-6-solving-systems-with-gaussian-elimination), explains augmented matrices and elementary row operations. The book is CC BY 4.0.
How to use it
- Enter one equation per row as coefficients followed by its right-hand side.
- Use the same number of equations and unknowns, from two to four; decimals and fractions are allowed.
- Read exact variable values or the system status, then open the elimination steps.
Privacy & limitations
Equations are solved in your browser and are not uploaded.
Related tools
Frequently asked questions
How do I enter a three-variable system?
Enter four values per row: the coefficients of x₁, x₂ and x₃, followed by the right-hand side. Enter three rows. For example, 2 1 -1 8 means 2x₁ + x₂ − x₃ = 8. Fractions such as -1/2 are accepted.
What does no solution mean?
The equations contradict each other. During elimination, a row with zero coefficients and a nonzero right-hand side means 0 equals a nonzero value. No choice of the unknowns can satisfy that row.
What does infinitely many solutions mean?
Some equations repeat information already present in the others. One or more variables remain free, and the leading variables are written in terms of them. The free-variable labels and elimination steps help explain the family of solutions.
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