System of Equations Solver
Solve a system of 2 or 3 linear equations using Cramer's rule - enter the coefficients and get an exact solution instantly.
x =
3
y =
3
Recommended Settings
Pro Tips
- •Cramer's rule solves a system by comparing determinants - it gives an exact answer in one step without the back-substitution needed in elimination
- •A determinant of zero for the full coefficient matrix means the system doesn't have exactly one solution - it's either inconsistent (no solution) or dependent (infinitely many)
- •Enter coefficients as they'd appear in the equation, including negative signs - a missing variable in an equation should be entered as 0
- •This solves exact linear systems - for word problems, translate each sentence into an equation first, then enter the coefficients here
Most Popular
Most students use the default 3-equation example to see the format before entering their own system
When to Use This Tool
Solve a system of 2 or 3 linear equations for exact values of each variable.
Get the solution after translating a word problem into a system of equations.
Verify an answer found manually using elimination or substitution.
See how Cramer's rule uses determinants to solve linear systems directly.
How It Works
Compute the determinant of the full coefficient matrix
For each variable, replace its column in the coefficient matrix with the constants column and compute that determinant
Divide each modified determinant by the original determinant to get the value of each variable, following Cramer's rule
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Lightning Fast
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Open Source
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Frequently Asked Questions
What does it mean if there's no unique solution?
It means the coefficient matrix's determinant is zero. This happens when the equations are dependent (one is a multiple of another, giving infinitely many solutions) or inconsistent (contradictory, giving no solution).
How do I enter an equation missing a variable?
Enter 0 as that variable's coefficient - for example, the equation 2x + 5 = 11 (with no y term) in a 2-variable system would be entered as coefficients 2 and 0, with constant 11.
What's Cramer's rule?
It's a method for solving linear systems using determinants - each variable's value is the ratio of two determinants, one from a modified coefficient matrix and one from the original.
Is my data sent anywhere?
No. All calculations happen locally in your browser.