Calculators/Mathematics/System of linear equations

System of linear equations

Determinant method (Cramer's rule) · 2×2 and 3×3

System of linear equations — illustration
x+y=
x+y=
Solution (unique solution)
x = 2
y = 1
Determinants (Cramer's rule)
W = -7
Wₓ = -14
W_y = -7
Cramer's formulas: x = Wₓ/W, y = W_y/W

Solves systems of linear equations with the determinant method (Cramer's rule). For a 2×22\times2 system we compute the main determinant and the numerator determinants:

W=∣a1b1a2b2∣x=WxW, y=WyWW=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}\quad x=\frac{W_x}{W},\ y=\frac{W_y}{W}

The type of system depends on the determinants: W≠0W\neq0 → a unique solution; W=0W=0 with a nonzero numerator → inconsistent (no solutions); all determinants =0=0 → dependent (infinitely many). The calculator also handles 3×33\times3 systems.

Examples

2x + y = 5, x − 3y = −1
→ W = −7 → x = 2, y = 1
x + y = 2, 2x + 2y = 5
→ W = 0, Wₓ ≠ 0 → inconsistent system (parallel lines)

Frequently asked questions

When is a system inconsistent and when is it dependent?+

When W = 0: if any numerator determinant is ≠ 0 the system is inconsistent (no solutions, parallel lines). If all numerators are also = 0 it is dependent (the lines coincide).

How does this differ from the simulation in the Laboratory?+

The simulation shows geometrically how the position of two lines determines the number of solutions. This calculator computes the exact solution (including 3×3) by Cramer's rule.

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