What method does the solver use?

For 2-variable systems it applies Cramer's rule, computing the determinant and two sub-determinants to find x and y directly. For 3-variable systems it uses Gaussian elimination with partial pivoting, which is numerically more stable than Cramer's rule for larger systems.

What happens if there is no unique solution?

The solver detects two special cases. If the determinant (or the reduced row echelon rank) is zero and the equations are consistent, the system has infinitely many solutions (the lines or planes coincide). If the system is inconsistent the lines or planes are parallel and there is no solution. Both cases are reported clearly.

How accurate are the results?

All arithmetic uses JavaScript 64-bit floating point (IEEE 754 double precision), which gives about 15-16 significant decimal digits. For typical classroom or engineering coefficients the answers are exact to at least 10 decimal places. Ill-conditioned systems with near-zero determinants may show small rounding artefacts.

What does the diagram show for 2-variable systems?

Each equation ax + by = c is a straight line in the xy-plane. The blue line represents equation 1 and the orange line represents equation 2. The red dot marks the unique solution — the point where both lines cross. The window is centred on the solution with a span of plus or minus 4 units.

Can I solve systems with fractional or negative coefficients?

Yes. All coefficient fields accept any real number including decimals and negatives. For example entering -0.5 or 3.14159 works just as well as whole numbers.

Is there a 4x4 or larger solver?

Not currently. For most coursework and practical problems 2x2 and 3x3 cover the vast majority of cases. Larger systems are best handled by dedicated matrix software such as NumPy, MATLAB, or Wolfram Alpha where you can paste the full matrix.

System of Equations Solver

A system of linear equations is a set of two or more equations that share the same unknowns. Solving the system means finding the values of those unknowns that satisfy every equation simultaneously. This tool handles the two most common cases — two equations in two unknowns (2x2) and three equations in three unknowns (3x3) — and shows you the complete solution together with the reasoning behind every step.

How it works

2-variable systems — Cramer’s rule

For a 2x2 system written as:

a1·x + b1·y = c1
a2·x + b2·y = c2

the solution uses the determinant of the coefficient matrix:

D  = a1·b2 - b1·a2
Dx = c1·b2 - b1·c2
Dy = a1·c2 - c1·a2

x = Dx / D
y = Dy / D

If D = 0 the equations are either parallel (no solution) or the same line (infinitely many solutions). The tool detects both cases and reports them.

3-variable systems — Gaussian elimination

For three equations the solver builds the augmented matrix and applies forward elimination with partial pivoting — at each step it swaps rows to place the largest available pivot on the diagonal. This avoids division by near-zero values, keeping the arithmetic stable. After three elimination passes, back-substitution recovers z, then y, then x.

Geometric meaning (2x2)

Every linear equation in two variables is a straight line. Solving the system geometrically means finding where the two lines cross. The embedded diagram plots both lines with the intersection point highlighted in red, making it easy to see whether the system has one solution (lines cross at a point), no solution (lines are parallel), or infinitely many (lines are identical).

Worked example — 2 variables

Consider the default system:

2x + 3y = 8
5x -  y = -7

Step 1: compute the determinant.

D = (2)(-1) - (3)(5) = -2 - 15 = -17

Step 2: apply Cramer’s rule.

Dx = (8)(-1) - (3)(-7) = -8 + 21 = 13
Dy = (2)(-7) - (8)(5)  = -14 - 40 = -54

x = 13 / (-17) ≈ -0.7647
y = -54 / (-17) ≈  3.1765

Step 3: verify by substituting back.

Eq 1: 2(-0.7647) + 3(3.1765) = -1.5294 + 9.5294 = 8.0000  ✓
Eq 2: 5(-0.7647) + (-1)(3.1765) = -3.8235 - 3.1765 = -7.0000  ✓

Worked example — 3 variables

Consider the classic 3x3 system:

 2x +  y -  z =  8
-3x -  y + 2z = -11
-2x +  y + 2z = -3

Gaussian elimination with partial pivoting produces the row echelon form:

[ -3   -1    2  | -11 ]
[  0  0.33  0.67|  0.67]
[  0    0    1  | -1  ]

Back-substitution gives z = -1, y = 3, x = 2. Substituting back confirms all three original equations are satisfied exactly.

Formula reference

System	Method	Key formula
2x2	Cramer’s rule	x = Dx/D, y = Dy/D
3x3	Gaussian elim.	forward sweep then back-sub
Singular (D=0)	—	no unique solution

Every computation runs entirely inside your browser — no coefficients are sent to a server.