Calcolatrice di sistemi di equazioni
Questa calcolatrice risolve due equazioni in x e y, oppure tre in x, y e z, e dà ogni risposta come frazione esatta. Scegli quante equazioni hai, inserisci i numeri e premi Calcola.
Risultato
Inserisci i valori e premi Calcola per vedere il risultato.
Formula
Cramer’s rule for two equations, valid when the determinant a₁b₂ − a₂b₁ is not zero.
For three equations, D is the determinant of the 3×3 grid of coefficients, and Dₓ, D_y, D_z are the same determinant with one column replaced by the right-hand sides.
Come usare questa calcolatrice
- Choose two equations (x and y) or three equations (x, y and z).
- Write every equation with the variables in the same order and the number on the right: ax + by = c, or ax + by + cz = d. A missing variable has coefficient 0.
- Enter the coefficients — whole numbers, decimals or fractions such as 3/4 — and press Calculate.
- Read the answer first, then the steps: the determinant, Cramer’s rule and a check in every equation.
Esempi svolti
Easy: solve 2x + 3y = 8 and x − y = −1
Enter a₁ = 2, b₁ = 3, c₁ = 8 and a₂ = 1, b₂ = −1, c₂ = −1 (x − y = −1 means the y coefficient and the right side are both −1).
Determinant: D = 2·(−1) − 1·3 = −5. It is not 0, so there is exactly one solution.
x = (8·(−1) − (−1)·3) ÷ (−5) = (−5) ÷ (−5) = 1 and y = (2·(−1) − 1·8) ÷ (−5) = (−10) ÷ (−5) = 2.
Check: 2·1 + 3·2 = 8 ✓ and 1 − 2 = −1 ✓. Solving by elimination (multiply the second equation by 3 and add) gives the same x = 1, y = 2.
Medium: a 3 variable system
Solve x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2. Choose three equations and enter the rows 1, 1, 1, 6 / 2, −1, 1, 3 / 1, 2, −1, 2.
The coefficient determinant is D = 7, so there is exactly one solution.
Replacing each column by the right-hand sides gives Dₓ = 7, D_y = 14 and D_z = 21, so x = 7/7 = 1, y = 14/7 = 2 and z = 21/7 = 3.
Check: 1 + 2 + 3 = 6 ✓, 2 − 2 + 3 = 3 ✓, 1 + 4 − 3 = 2 ✓.
Hard: a system with fraction answers and a missing variable
Solve 2x + y = 1, y + z = 1 and x + z = 1. Each equation is missing one variable, so enter 0 for it: 2, 1, 0, 1 / 0, 1, 1, 1 / 1, 0, 1, 1.
D = 3, Dₓ = 1, D_y = 1 and D_z = 2, so x = 1/3, y = 1/3 and z = 2/3 — exact fractions, not 0.33 and 0.67.
Check: 2·(1/3) + 1/3 = 1 ✓, 1/3 + 2/3 = 1 ✓, 1/3 + 2/3 = 1 ✓.
Errori comuni
- Leaving a gap for a missing variable. In y + z = 1 the x coefficient is 0 — enter 0, don’t skip it or shift the numbers along.
- Dropping a minus sign. x − y = −1 has a y coefficient of −1 and a right side of −1.
- Writing the variables in a different order in different equations. Keep x, then y, then z in every row.
- Reading D = 0 as “no solution”. A zero determinant means either no solution or infinitely many; Sumzo checks which.
- Rounding to decimals halfway through. Keep fractions such as 1/3 until the end, then check the answer in every equation.
Domande frequenti
How many solutions can a system of equations have?
Exactly one, none, or infinitely many. Two lines either cross once, are parallel (no solution) or are the same line (infinitely many). Three equations in x, y and z behave the same way with planes. The calculator tells you which case you have.
Can this calculator solve a 3 variable system of equations?
Yes. Choose “Three equations (x, y and z)” and enter the four numbers of each equation. Larger systems, or equations with x², xy or other non-linear terms, can go to the Sumzo AI solver on the homepage.
Is this the same as solving simultaneous equations by elimination or substitution?
It finds the same answer. Elimination, substitution and Cramer’s rule are different routes to one solution; the steps here use Cramer’s rule because it also shows straight away whether there is one solution.
What if the determinant is zero?
Then there is no single solution. With two equations the lines are parallel (no solution) or the same line (infinitely many solutions); with three, the equations contradict each other or one of them adds no new information. Sumzo tells you which.
What makes a system linear?
Every variable appears on its own, to the first power, multiplied only by a number — like 2x + 3y = 8. A linear system calculator like this one cannot solve x² + y = 5 or xy = 6.
Can I enter fractions and decimals?
Yes — 3/4, −2/5 or 0.5 all work. Sumzo works with exact fractions, so 1/3 stays 1/3 instead of turning into 0.333….