Systems of Equations Solver

Cramer's Rule Calculator — Step-by-step determinants for 2×2 and 3×3 linear systems.

Cramer's Rule in a Nutshell

For a 2×2 system:

\( a_1 x + b_1 y = c_1 \)

\( a_2 x + b_2 y = c_2 \)

Determinants: \[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix},\; D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix},\; D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \]

Solution: \[ x = \frac{D_x}{D} ,\; y = \frac{D_y}{D} \quad (D\neq0) \].

For a 3×3 system :

\( a_1 x + b_1 y + c_1 z = d_1 \)

\( a_2 x + b_2 y + c_2 z = d_2 \)

\( a_3 x + b_3 y + c_3 z = d_3 \)

Determinants: \[ D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix},\; D_x = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix},\; D_y = \begin{vmatrix} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \end{vmatrix},\; D_z = \begin{vmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \end{vmatrix} \]

Solution: \[ x = \frac{D_x}{D} ,\; y = \frac{D_y}{D},\; z = \frac{D_z}{D} \quad (D\neq0) \].

Solve 2×2 Linear System

Cramer's rule with step‑by‑step determinants

\( 1x + 2y = 3 \)

\( 1x + 6y = 5 \)

Eq 1 :
Eq 2 :
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Solve 3×3 Linear System

Cramer's rule with full determinant expansion

\( 1x + 2y + (-3)z = 9 \)

\( 1x + 6y + 5z = -1 \)

\( 1x + 6y + 4z = 3 \)

Eq 1 :
Eq 2 :
Eq 3 :
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More Resources

analyzemath.com — step‑by‑step linear system solver using Cramer's rule.