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System of Equations Solver Cramer's rule.

Solve a linear system of equations with 2 or 3 unknowns — determinant, solution and special cases shown instantly

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System of Equations Solver

Solve a linear system of equations with 2 or 3 unknowns — using Cramer's rule

2x + 3y = 12

1x − 1y = 1

x +y =
x +y =
x

3

y

2

Determinant D = -5

Cramer's rule:

x_i = D_i / D

D = determinant of the coefficient matrix

Cramer's rule – step by step

Solving linear systems of equations with determinants

A linear system of equations with two or three unknowns often comes up in school, university and engineering — for example in mixture problems, motion problems, or when solving equilibrium conditions in physics. Cramer's rule solves such systems using determinants: x_i = D_i / D. Here D is the determinant of the coefficient matrix and D_i is the determinant in which the i-th column has been replaced by the right-hand side.

The decisive special case is D = 0: here there is no unique solution. To distinguish whether the system has infinitely many solutions (the lines or planes coincide) or no solution at all (the lines or planes are parallel but distinct), you check the numerator determinants Dx, Dy (and, for three unknowns, Dz): if these are all also 0, the system is underdetermined. If at least one of them is nonzero, the system is inconsistent.

Calculation example: 2 unknowns

2x + 3y = 12 ; x − y = 1

2x + 3y = 12 ; x − y = 1
ItemAmount
D = a1×b2 − a2×b1-5 (= 2×(−1) − 1×3)
Dx = c1×b2 − c2×b1−15 (= 12×(−1) − 1×3)
Dy = a1×c2 − a2×c1−10 (= 2×1 − 1×12)
x = Dx / D3 (= −15 / −5)
y = Dy / D2 (= −10 / −5)

Calculation example: 3 unknowns

x+y+z=6 ; 2y+5z=−4 ; 2x+5y−z=27

x+y+z=6 ; 2y+5z=−4 ; 2x+5y−z=27
ItemAmount
Determinant D-21
x = Dx / D5 (= −105 / −21)
y = Dy / D3 (= −63 / −21)
z = Dz / D-2 (= 42 / −21)

Frequently asked questions about the system of equations solver

Cramer's rule, determinants and special cases explained

Cramer's rule solves a linear system of equations using determinants: x_i = D_i / D, where D is the determinant of the coefficient matrix and D_i is the determinant in which the i-th column has been replaced by the right-hand side (the constants). It works for any number of unknowns, but is mainly practical for small systems (2–3 unknowns), since computing determinants quickly becomes more work as the system grows.

If D = 0, there is no unique solution. There are two cases: if all the D_i (Dx, Dy, and Dz if applicable) are also 0, the system is underdetermined and has infinitely many solutions (the lines or planes coincide). If at least one D_i ≠ 0, the system is inconsistent and has no solution (the lines or planes run parallel without intersecting).

Coefficient matrix: D = 2×(−1) − 1×3 = −2−3 = −5. Dx (first column replaced by the constants): Dx = 12×(−1) − 1×3 = −12−3 = −15. Dy = 2×1 − 1×12 = 2−12 = −10. This gives x = Dx/D = −15/−5 = 3 and y = Dy/D = −10/−5 = 2. Check: 2×3+3×2=12 ✓ and 3−2=1 ✓.

Yes. With 3 unknowns (x, y, z), you work with 3×3 determinants: D, Dx, Dy and Dz. The principle stays the same — each variable is given by x_i = D_i / D. Classic example: x+y+z=6, 2y+5z=−4, 2x+5y−z=27 has the solution x=5, y=3, z=−2.

For 2 or 3 unknowns, Cramer's rule is very clear and quick to follow by hand. For larger systems (4 or more unknowns), computing determinants becomes very laborious — Gaussian elimination is more efficient there.

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