Solve Simultaneous Linear Equations
This system of equations solver finds the values of the unknowns in a set of linear equations with 2 or 3 variables. Enter the coefficients for each row and it applies Cramer's Rule instantly, showing every determinant behind the answer.
Who It's For
Algebra & Homework
Word Problems
Linear Algebra Practice
How to Solve a System of Equations
Choose the System Size
Select 2 variables (2×2) or 3 variables (3×3) mode to match your problem.
Enter the Coefficients
Fill in the coefficients and constant for each equation row. Empty fields are treated as 0.
Read the Solution
The values of x, y (and z for 3×3) calculate automatically as you type, shown as fractions when applicable.
Follow the Determinants
Read the step-by-step view to see the main determinant and each variable's determinant computed in full.
What Cramer's Rule Does
Cramer's Rule expresses each variable as the ratio of two determinants. Divide the variable's determinant by the main determinant D:
D ≠ 0.Each Dx, Dy, Dz is found by replacing that variable's column in the coefficient matrix with the column of constants.
Worked Examples
| System | Determinants | Result |
|---|---|---|
| x + 2y = 5 3x − y = 1 | D = −7, Dx = −7, Dy = −14 | x = 1, y = 2 |
| 2x + y = 5 4x + 2y = 10 | D = 0, all Dᵢ = 0 | ∞ solutions |
| x + y = 1 x + y = 2 | D = 0, some Dᵢ ≠ 0 | No solution |
Features
2×2 and 3×3 Systems
Toggle freely between two-variable and three-variable systems.
Cramer's Rule with Determinants
Solves with the determinant method, the same approach taught in linear algebra courses.
Step-by-Step Solutions
Numbered steps show the main determinant and each variable's determinant computed in full.
Special Cases
Detects dependent (infinite solutions) and inconsistent (no solution) systems.
Exact Fraction Results
Displays solutions as exact fractions whenever the values are not whole numbers.
Instant Results
Solves live as you type — no submit button, with empty fields counted as 0.
How the Main Determinant Decides the Outcome
| Outcome | Condition | Meaning |
|---|---|---|
| Unique solution | D ≠ 0 | Exactly one set of values satisfies the system. |
| Infinite solutions | D = 0 and all Dᵢ = 0 | The equations are dependent — they describe the same relationship. |
| No solution | D = 0 and some Dᵢ ≠ 0 | The equations are inconsistent — they contradict each other. |
Frequently Asked Questions
What is Cramer's Rule?
Cramer's Rule is a method that uses determinants to solve systems of linear equations. Each variable equals the ratio of two determinants: its own determinant divided by the main determinant D.
Can it solve 2 and 3 variable systems?
Yes. Use the toggle to choose 2-variable (2×2) or 3-variable (3×3) mode. The grid of coefficient inputs adjusts automatically and the solution recalculates instantly.
When does a system have no solution?
When the main determinant D = 0 and at least one of Dx, Dy (or Dz) is non-zero. The equations contradict each other, so no set of values can satisfy them all.
When does it have infinite solutions?
When all determinants — D, Dx, Dy (and Dz) — equal 0. The equations are dependent, meaning they overlap, so infinitely many value sets satisfy the system.
What is a determinant?
A determinant is a single number computed from a square matrix of coefficients. In Cramer's Rule, the determinants encode whether the system has a unique, infinite, or no solution and yield the values directly.
Does it show the steps?
Yes. The solver lists numbered steps: it writes out the system, computes the main determinant D, then each variable's determinant by column replacement, and finally the result.
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