System of Equations Calculator
Type your equations the way you’d write them — 2x + 3y = 7, x − y = 1 — and the calculator solves the system exactly, using fractions rather than rounded decimals. It tells you whether there is one solution, no solution or infinitely many, writes the general solution when there are free variables, and shows each row operation of Gaussian elimination.
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Linear system solver
Use any letters as variables (x, y, z, a, b, x1…). Coefficients can be integers, decimals or fractions such as 1/2x or x/3. Terms may appear on both sides.
Show elimination steps
Linear Systems Worksheet Pack
Printable systems of equations practice: 2×2 and 3×3 worksheets with full answer keys, a method comparison guide and word-problem set.
- 2×2 worksheet (PDF, DOCX)
- 3×3 worksheet (PDF, DOCX)
- Word problems (PDF, DOCX)
- Methods guide (PDF)
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Printable Methods for Solving Systems GuideWhat is a system of linear equations?
A system of linear equations is a set of equations that share the same unknowns, where each unknown appears only to the first power — no squares, products of variables or square roots. Solving the system means finding values that make every equation true at once. With two unknowns, each equation is a straight line on a graph and the solution is where the lines cross; with three unknowns, each equation is a flat plane in space.
One, none or infinitely many solutions
| Case | What it looks like | Graph (2 unknowns) |
|---|---|---|
| One solution | Elimination gives a leading 1 for every variable | Lines cross at a single point |
| No solution | A row becomes 0 = non-zero number | Parallel lines that never meet |
| Infinitely many | A row becomes 0 = 0 and some variables are free | The same line written twice |
A linear system can never have exactly two or three solutions — if it has more than one, it has infinitely many.
How Gaussian elimination works
- Write the coefficients and constants as an augmented matrix, one row per equation.
- Pick a pivot — a non-zero entry in the first column — and swap it to the top if needed.
- Divide the pivot row so the pivot becomes 1.
- Subtract multiples of the pivot row from every other row to make zeros above and below the pivot.
- Move to the next column and repeat.
- Read the solution from the reduced matrix, or spot an impossible row.
This full reduction (Gauss–Jordan elimination) is what the calculator shows in its steps. It uses exact fractions throughout, so there are no rounding errors.
Worked example
Solve 2x + 3y − z = 5, x − y + 2z = 3 and 3x + y + z = 10. After eliminating x from the second and third rows and then y, the matrix reduces to x = 23/5, y = −2 and z = −9/5 — you can check that 2(23/5) + 3(−2) − (−9/5) = 46/5 − 6 + 9/5 = 5. The calculator gives the exact fractions and their decimal values (4.6, −2 and −1.8).
Other methods
- Substitution: solve one equation for a variable and substitute into the others — quick for 2×2 systems with a coefficient of 1.
- Elimination (addition): add or subtract multiples of equations to cancel a variable — the hand version of what the calculator does.
- Cramer’s rule: each variable is a ratio of determinants; neat for 2×2 and 3×3 but slow for larger systems.
- Matrix inverse: x = A⁻¹b when A is square and invertible.
- Graphing: plot the lines and find the intersection — good for intuition, less precise.
Writing equations for the calculator
| You can type | Means |
|---|---|
| 2x + 3y = 7 | Standard form |
| y = 2x - 1 | Variables on the right side are moved across |
| 1/2x + y/4 = 3 | Fractional coefficients |
| 0.5a - 1.25b = 2 | Decimals and any variable letters |
| x1 + x2 + x3 = 1 | Subscript-style variable names |
Brackets and products of variables such as xy are not supported, because they are not linear — expand them first.
Solving a 2×2 system by hand
- Write both equations in the form ax + by = c.
- Choose a variable to eliminate and multiply the equations so its coefficients match.
- Subtract (or add) the equations to remove that variable.
- Solve the single equation that remains.
- Substitute back to find the other variable.
- Check both original equations.
For 2x + 3y = 7 and x − y = 1: multiply the second by 3 to get 3x − 3y = 3, add to the first to get 5x = 10, so x = 2 and then y = 1.
Checking your answer
Always substitute the solution back into every original equation — both sides should match exactly. The calculator works in exact fractions, so its results pass this check without rounding error; if your hand answer differs, compare your row operations with the steps shown.
Determinants and unique solutions
A square system has exactly one solution when the determinant of its coefficient matrix is not zero. For a 2×2 system ax + by = e, cx + dy = f, the determinant is ad − bc. If it is zero, the lines are parallel or identical, so there is either no solution or infinitely many — the calculator tells you which by checking the constants during elimination.
Where linear systems are used
Linear systems appear in mixture problems (how much of two solutions to combine), in balancing chemical equations, in electrical circuits using Kirchhoff’s laws, in economics input–output models, in computer graphics and in fitting a line to data. Many real problems have hundreds or thousands of unknowns and are solved by computers with the same elimination ideas.
Privacy
The solver runs entirely in your browser; your equations are not uploaded.
Frequently asked questions
How do I solve a system of 3 equations with 3 variables?
Type each equation on its own line; the calculator eliminates variables step by step and gives exact values.
What does “no solution” mean?
The equations contradict each other — for example, x + y = 2 and x + y = 5.
What does “infinitely many solutions” mean?
At least one equation is a combination of the others, so some variables can take any value.
Can I use fractions?
Yes, like 1/2x or x/3, and answers are given as exact fractions.
How many equations can it solve?
Up to six equations with up to six unknowns.
Does it show steps?
Yes, open “Show elimination steps”.
What if I have more equations than unknowns?
The calculator still solves it; extra equations must be consistent or it reports no solution.