System of 3 linear equations solver
Write each equation as ax + by + cz = d and enter the twelve numbers. Handy for linear algebra exercises and mixture or allocation problems.
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How to use it
- Enter a1, b1, c1, d1 (constant 1), a2 and b2.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
Compute the coefficient determinant D, replace each unknown's column with the constants to get Dx, Dy, Dz, then x = Dx/D, y = Dy/D, z = Dz/D. If D = 0 the ranks of the matrix and augmented matrix decide the case.
Example
Inputs: a1 = 2, b1 = 1, c1 = -1, d1 (constant 1) = 8, a2 = -3. Result: Solution: x = 2, y = 3, z = -1; x: 2; y: 3.
Limits and caveats
- Linear equations only.
- When the determinant is extremely close to zero, decimal accuracy can suffer.
Questions about this tool
What if an unknown is missing from an equation?
Enter 0 as its coefficient. For 2y + 5z = -4 use a = 0, b = 2, c = 5.
Why Cramer's rule?
It gives each unknown directly from determinants and flags the special case when the determinant is zero.
How can I check the answer?
Substitute x, y and z into all three equations; each left side should equal its constant.