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Linear System 3x3 Solver

Solve a 3x3 linear system (three equations, three unknowns) with Cramer's rule and 3x3 determinants.

Linear System 3x3

Enter all three equations. Results update live.

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

Result
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Enter numbers to see the result.

x
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y
-
z
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Formula: Dx, Dy, Dz are determinants with the constants column swapped in. x = Dx/D, y = Dy/D, z = Dz/D.
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Cramer's rule for 3x3

For three equations, Cramer's rule builds four 3x3 determinants: the main one from the coefficients, plus Dx, Dy, and Dz where the constants column replaces the x, y, or z column. Each variable is its determinant divided by the main determinant.

Checking your answer

Plug the solution back into all three equations: each left side should equal its right side. With the default example the answer is x = 5, y = 3, z = -2, which satisfies every equation.

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Linear System 3x3 FAQs

How do I solve x + y + z = 6, 2y + 5z = -4, 2x + 5y - z = 27?
Enter the coefficients row by row as shown in the preloaded example. The solution is x = 5, y = 3, z = -2.
What is a 3x3 determinant?
It is a single number computed from a 3x3 grid of coefficients. A non-zero determinant means the system has exactly one solution; a zero determinant means it does not.
What if the determinant is zero?
Then the equations are dependent or contradictory, so there is no unique solution. The tool tells you instead of producing a wrong answer.
Can the coefficients be decimals?
Yes, any real numbers work. Negative values and decimals are all fine.
Is my system uploaded to a server?
No. All the determinant math runs locally in your browser.
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