System of Equations Solver
Solve 2 or 3 linear equations exactly, with Gauss-Jordan or Cramer’s rule steps and a graph of two lines. Says when there is no solution or infinitely many. How it works
Use the System of Equations Solver
About the System of Equations Solver
2x + 3y = 18 and 4x − y = 8 have the solution x = 3, y = 4. This system of equations solver takes 2 equations in x and y or 3 in x, y and z, solves them by exact elimination so fractions stay exact, and says when there is no solution or infinitely many. On request it shows the Gauss-Jordan row operations or Cramer’s rule, and it draws a 2 by 2 system as two lines.
Worked systems and what the tool returns
| Equations | Result |
|---|---|
| 2x + 3y = 18; 4x − y = 8 | x = 3, y = 4 |
| x + y = 10; x − y = 2 | x = 6, y = 4 |
| x + y = 10; 12x + 8y = 104 | x = 6, y = 4 |
| 3x + 2y = 7; x − 4y = 0 | x = 2, y = 0.5 (1/2) |
| 0.5x + 0.25y = 3; x + y = 8 | x = 4, y = 4 |
| x + 2y = 3; 2x + 4y = 6 | Infinitely many: x = 3 − 2y, y = any value |
| x + 2y = 3; 2x + 4y = 7 | No solution |
| x + y + z = 6; 2y + 5z = −4; 2x + 5y − z = 27 | x = 5, y = 3, z = −2 |
| 2x + y − z = 8; −3x − y + 2z = −11; −2x + y + 2z = −3 | x = 2, y = 3, z = −1 |
| x + y + z = 6; x + 2y + 3z = 14; 2x + 3y + 4z = 20 | Infinitely many: x = −2 + z, y = 8 − 2z, z = any value |
| x + 2y + 3z = 1; 4x + 5y + 6z = 2; 7x + 8y + 9z = 4 | No solution |
In the tool, type each coefficient in its own box: 4x − y = 8 is 4, -1 and 8.
Frequently asked questions
- How do you solve a system of equations step by step?
- By elimination: for 2x + 3y = 18 and 4x − y = 8, multiply the second equation by 3 to get 12x − 3y = 24, then add it to the first: 14x = 42, so x = 3. Substitute back: y = 4x − 8 = 4. The tool reduces the equations the same way and shows x = 3, y = 4; press Show steps under Elimination steps to see each row operation and the matrix after it.
- How do you know if a system has no solution or infinitely many?
- Reduce it. x + 2y = 3 and 2x + 4y = 7 reduce to 0 = 1, a contradiction, so there is no solution. x + 2y = 3 and 2x + 4y = 6 are the same line, so there are infinitely many: x = 3 − 2y for any y. The tool decides by comparing the rank of the coefficients with the rank of the whole system.
- Can it solve 3 equations with 3 unknowns?
- Yes. Switch to 3x3. For x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27 it gives x = 5, y = 3, z = −2. Check: 5 + 3 − 2 = 6.
- How does Cramer’s rule use the determinant?
- x = det(Aₓ)/det(A), where Aₓ has the constants in place of the x column. For 2x + 3y = 18 and 4x − y = 8, det(A) = 2×(-1) − 3×4 = -14, det(Aₓ) = 18×(-1) − 3×8 = -42 and det(Aᵧ) = 2×8 − 18×4 = -56, so x = -42/-14 = 3 and y = -56/-14 = 4. Press Show steps and pick Cramer’s rule to see each replaced matrix and its determinant. When det(A) = 0 the rule does not apply; the tool says so and points to Gauss-Jordan, which tells no solution from infinitely many.
- Can I enter decimals, and are fractional answers exact?
- Decimals are read exactly, so 0.5x + 0.25y = 3 with x + y = 8 gives x = 4, y = 4. A fractional answer is shown as a decimal and a fraction: 3x + 2y = 7 and x − 4y = 0 give y = 0.5 (1/2).
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