Using the System of Equations Calculator
Use two equations to find two unknowns, such as item prices or the intersection of straight lines. Write each equation in ax + by = c form first. The second row uses d, e and f to avoid confusing the coefficients.
Worked example
Use the example inputs below to reproduce this result. These are the form's starting values, not recommended targets. Changing your inputs updates your result above; this worked example stays fixed for comparison. Results are rounded for display.
Example inputs
- a in ax + by = c
- 2
- b in ax + by = c
- 1
- c in ax + by = c
- 7
- d in dx + ey = f
- 1
- e in dx + ey = f
- -1
- f in dx + ey = f
- 2
Example result
- Solution
- x = 3, y = 1
- x
- 3
- y
- 1
- Verification
- Exact cross-products satisfy both original equations.
- Steps
- Start with 2x + (1)y = 7 and 1x + (-1)y = 2. Compute D = a·e − b·d = -3. Compute Dx = c·e − b·f = -9; Dy = a·f − c·d = -3. Divide: x = Dx/D = 3; y = Dy/D = 1.
Method and assumptions
Use 2×2 determinants (Cramer’s rule). If the determinant is zero, compare augmented constraints, including zero rows, to distinguish infinitely many solutions from no solution. Two equations and integer coefficients only.
Reference: openstax.org — method and further reading
Check the answer in the original problem
After solving, substitute the result into the original equation or relationship where possible. This can catch a sign error or a coefficient entered in the wrong position. Keep the exact inputs available when comparing with another tool, since changing a coefficient or rounding an intermediate value can change the problem being solved.
Common question
What does a zero determinant mean?
There is no unique intersection. Compatible dependent equations have infinitely many solutions, while contradictory constraints have none. Zero rows are checked separately.