Question 1

How should I type the equations?

Accepted Answer

One equation per line, with an equals sign separating the two sides, for example "2x + 3y = 8" or "x - y = -1". The calculator accepts an implicit coefficient ("x" means the same as "1x"), a negative sign, unknowns with any name, a variable appearing on both sides of the equals sign ("2x + 1 = x + 4"), and a variable that simply doesn't appear in one of the equations, it is treated as a zero coefficient automatically.

Question 2

What's the difference between the Gaussian elimination and Cramer's rule shown here?

Accepted Answer

They are two different methods for the same problem, shown side by side so you can compare them. Gaussian elimination transforms the augmented matrix step by step into reduced row echelon form, and works for any system, including ones without a single unique solution. Cramer's rule computes the solution directly from determinants, but only applies when the system is square (same number of equations as unknowns) and the coefficient matrix's determinant is not zero, when that isn't the case, the tool explains why Cramer's rule can't be used instead of just leaving that section out.

Question 3

What does a "consistent and dependent" system mean?

Accepted Answer

It means there are infinitely many solutions, not a vague one: the calculator identifies which variables are free (can take any value) and writes the remaining ones in terms of those, for example "x = 2 - y", instead of just saying "infinitely many solutions". Plugging in any value for the free variables gets you a valid solution to the system.

Question 4

How do I know a system is inconsistent, rather than that I typed something wrong?

Accepted Answer

When Gaussian elimination reaches a row like "0 = 5" (every coefficient zeroed out, but the constant term is not zero), that mathematically proves the equations contradict each other, there is no combination of values that satisfies all of them at once. The calculator shows exactly that inconsistent row as evidence, instead of just saying "no solution".

Question 5

Why does the answer show up as a fraction instead of a decimal number?

Accepted Answer

Because all the internal arithmetic uses exact fractions (integer numerator and denominator, reduced by their greatest common divisor), never floating point. That avoids the classic calculator noise, like 0.1 + 0.2 coming out as 0.30000000000000004, and guarantees that a solution like 5/7 stays exactly 5/7 after several elimination steps, instead of an accumulated approximation. When the decimal value is exact (a denominator with only factors of 2 and 5), it is shown alongside; when it isn't, it is marked with "≈" to flag the approximation.

Question 6

How many unknowns can a system have?

Accepted Answer

Up to 6 unknowns, and any number of equations. Systems with more equations than unknowns (overdetermined) or fewer (underdetermined) are solved normally by Gaussian elimination, Cramer's rule only comes into play when the number of equations equals the number of unknowns.

Question 7

What does the "ill-conditioned" system warning mean?

Accepted Answer

It shows up when the pivots used during elimination have very different magnitudes from one another, a sign that the system is sensitive: small changes to the original coefficients could shift the solution by a lot. The calculator flags that case explicitly instead of presenting the result as if it were just as reliable as a well-behaved system.

Question 8

Is my data saved on any server?

Accepted Answer

No. All the equation parsing and all the solving (Gaussian elimination, Cramer's rule, verification) happen in your browser, nothing is sent to any server.

Linear Equation System Solver