System of Equations Calculator
Calculate system of equations instantly in your browser.
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What this tool does
Calculate system of equations instantly in your browser. A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
How to use the System of Equations Calculator
- Enter or select system size.
- Enter or select eq 1: coefficient of x.
- Enter or select eq 1: coefficient of y.
- Enter or select eq 1: coefficient of z.
- Enter or select eq 1: right-hand side.
- Enter or select eq 2: coefficient of x.
- Enter or select eq 2: coefficient of y.
- Enter or select eq 2: coefficient of z.
- Enter or select eq 2: right-hand side.
- Enter or select eq 3: coefficient of x.
- Enter or select eq 3: coefficient of y.
- Enter or select eq 3: coefficient of z.
- Enter or select eq 3: right-hand side.
- Read the calculated result; change any measurement to compare alternatives.
Formula
Solved with Cramer's rule: replace one column of the coefficient matrix with the right-hand sides, and each unknown is that determinant divided by the coefficient determinant.
- size
- System size
- a1
- Eq 1: coefficient of x
- b1
- Eq 1: coefficient of y
- c1
- Eq 1: coefficient of z
- r1
- Eq 1: right-hand side
- a2
- Eq 2: coefficient of x
- b2
- Eq 2: coefficient of y
- c2
- Eq 2: coefficient of z
- r2
- Eq 2: right-hand side
- a3
- Eq 3: coefficient of x
- b3
- Eq 3: coefficient of y
- c3
- Eq 3: coefficient of z
- r3
- Eq 3: right-hand side
A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
Worked example
For system of equations calculator, the following measurements illustrate the exact method: System size: 2; Eq 1: coefficient of x: 2; Eq 1: coefficient of y: 3; Eq 1: coefficient of z: 0; Eq 1: right-hand side: 12; Eq 2: coefficient of x: 1; Eq 2: coefficient of y: -1; Eq 2: coefficient of z: 0; Eq 2: right-hand side: 1; Eq 3: coefficient of x: 1; Eq 3: coefficient of y: 1; Eq 3: coefficient of z: 1; Eq 3: right-hand side: 6.
Inputs
- System size2 × 2 (two equations)
- Eq 1: coefficient of x2
- Eq 1: coefficient of y3
- Eq 1: coefficient of z0
- Eq 1: right-hand side12
- Eq 2: coefficient of x1
- Eq 2: coefficient of y-1
- Eq 2: coefficient of z0
- Eq 2: right-hand side1
- Eq 3: coefficient of x1
- Eq 3: coefficient of y1
- Eq 3: coefficient of z1
- Eq 3: right-hand side6
Result
- x3
- y2
- Determinant-5
- Check (Eq 1 left side)12
Results explained
- x
- x from the formula above. A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
- y
- y from the formula above. A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
- Determinant
- Determinant from the formula above. A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
- Check (Eq 1 left side)
- Check (Eq 1 left side) from the formula above. A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
Frequently asked questions
Solved with Cramer's rule: replace one column of the coefficient matrix with the right-hand sides, and each unknown is that determinant divided by the coefficient determinant.
A unique solution exists only when the determinant is nonzero; parallel or identical equations give determinant zero.
Where mathematically meaningful, yes. Fields specify permitted ranges and undefined operations display an error.
Exact integer calculations use safe integers; decimal calculations use browser floating point, so small rounding differences can occur.
All values remain in your browser and are never uploaded.