How do I solve three simultaneous equations?

Systematically eliminate one unknown at a time. Subtract multiples of the first equation from the other two until they no longer contain x, then subtract a multiple of the second from the third until it no longer contains y. The third equation now has only z, which gives z directly, and substituting backwards gives y and then x. That is Gaussian elimination, and it is what this page does — the alternative methods all reduce to it.

What does it mean if the determinant is zero?

That the three planes do not meet at a single point — and nothing more than that. It does not mean there is no solution. A zero determinant is consistent with infinitely many solutions (the planes meet along a line, or coincide) and with none at all (they are parallel, or form a prism). Distinguishing the two needs the constants, not just the coefficients, which is why this page compares ranks rather than checking a determinant.

How do I tell no solution from infinitely many?

Compare the rank of the coefficient matrix with the rank of the augmented matrix — the coefficients with the constants attached. If they are equal, the equations are consistent and there are infinitely many solutions. If the augmented rank is higher, elimination has produced a row saying zero equals something non-zero, and there is no solution. The same coefficients give either answer depending only on the constants: with 6, 12, 3 there is a line of solutions, and with 6, 13, 3 there are none.

Should I use Cramer’s rule instead?

It is fine for a 3×3 by hand and a poor choice generally. Cramer’s rule needs four determinants where elimination needs one pass, it becomes catastrophically slow as the system grows, and it says nothing useful when the determinant is zero — precisely the case that needs explaining. It also loses accuracy faster on near-singular systems. For learning what a determinant means it is instructive; for actually solving, elimination is what every numerical library uses.

Why does the solver swap rows around?

To divide by the largest available number rather than the first one to hand. Every elimination step divides by the pivot, and dividing by something small magnifies whatever rounding error is already present. Choosing the largest entry in the column — partial pivoting — keeps that amplification under control. On paper with exact fractions it makes no difference; in floating-point arithmetic it is the difference between a reliable answer and an unreliable one.

What is the residual and why does it matter?

It is what you get by putting the answer back into the original equations and seeing how far each one misses. A residual around 10⁻¹⁶ is floating-point zero and means the answer genuinely satisfies the system. The reason it is worth showing is that it tests against the equations you typed, not against the reduced ones the solver worked with — so it catches a mistyped coefficient, which no amount of correct arithmetic downstream would reveal.

What does it mean for a system to be ill-conditioned?

That it is solvable in principle and its answer is hypersensitive to the inputs. Geometrically, the three planes meet at a very shallow angle, so moving one of them slightly slides the intersection point a long way. A worked case: with the third equation 7x + 8y + 9.001z = 24 the answer is (0, 3, 0); change the 24 to 24.001 and it becomes (1, 1, 1); at 24.010 it is (10, −17, 10). A 0.04% change to one number moved the answer by orders of magnitude.

How do I know if my system is ill-conditioned?

The clearest practical signal is the spread of the pivots during elimination. If the largest is thousands of times the smallest, the system is close to singular and the answer is fragile — this page warns when that ratio passes ten thousand. A small residual does not rule it out: an ill-conditioned system can have an answer that satisfies the equations beautifully and still be worthless, because the equations themselves barely pin anything down.

What do three equations look like geometrically?

Each is a plane in three-dimensional space, and a solution is a point lying on all three. Three planes meeting at a single point is the ordinary case — one solution. They can also meet along a common line, or be the same plane three times, which gives infinitely many. Or they can fail to share any point at all: two parallel planes, or three planes arranged like the sides of a triangular prism, each pair meeting in a line but with no line common to all three.

Can I enter fractions or decimals?

Decimals, and a fraction should be converted first — 1/3 becomes 0.333333, with as many digits as you can give it. The arithmetic is done in floating point, so a recurring fraction is very slightly approximated, which is generally invisible but can matter for a system already close to singular. If your problem is exactly singular by design, entering 0.333 rather than 0.333333333 may be enough to make it look solvable when it is not.

Simultaneous Equations Solver (3×3)

Three equations, three unknowns, solved by Gaussian elimination with partial pivoting — and classified properly when there is no single answer. That last part is where most solvers give up. A zero determinant is usually reported as "no solution", which is wrong half the time: it means the system ha...

A zero determinant does not mean there is no solution. It means there is either no solution or an infinity of them, and only the constants decide which. This solver compares the rank of the coefficients with the rank of the full augmented system and says which case you have, rather than reporting a failure.

THE SYSTEM

x+y+z=

x+y+z=

x+y+z=

EXACTLY ONE SOLUTION

-1

Determinant -1, so the three planes meet at a single point. Substituted back into the equations as typed, the largest discrepancy is 8.88e-16 — which is rounding, not error.

Check	Left side	Right side	Difference
Equation 1	8	8	0
Equation 2	-11	-11	0
Equation 3	-3	-3	8.9e-16

THE WORKING

Rows are swapped to put the largest number on the diagonal. That is not tidiness — dividing by a small pivot amplifies rounding error, and choosing the largest available keeps the arithmetic stable. It is the difference between a method that works on paper and one that works on a machine.

Gaussian elimination with partial pivoting · live calculation

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter the nine coefficients and the three constants. A missing term is a zero — write it in rather than leaving the box empty.
2
Read the classification before the numbers. Exactly one solution, infinitely many, or none: the three cases mean very different things and only the first gives you values to use.
3
Check the verification table. It substitutes the answer back into the equations exactly as you typed them, which catches a mistyped coefficient in a way the solver itself cannot.
4
Take the ill-conditioning warning seriously if it appears. It means the arithmetic is fine and the answer is fragile, which is a different problem and a more dangerous one.

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Last updated: August 2, 2026 · Gaussian elimination with partial pivoting · A zero determinant means no solution or infinitely many, and the ranks decide which.