Unlock Linear Equations with Cramer's Rule Calculator

Visualize and solve systems of linear equations effortlessly! Enter your equations, and let our interactive tool guide you to the solution using Cramer's Rule.

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Choose the number of equations you need to solve.

System of Equations (2x2)

Equations: $$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$

Determinant Formulas (2x2)

$$ W = a_1b_2 - b_1a_2, \quad W_x = c_1b_2 - b_1c_2, \quad W_y = a_1c_2 - c_1a_2 $$

System of Equations (3x3)

Equation: $$ \begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases} $$

Determinant Formulas (3x3)

$$ W = a_1(b_2c_3 - c_2b_3) - b_1(a_2c_3 - c_2a_3) + c_1(a_2b_3 - b_2a_3) $$, $$ W_x = d_1(b_2c_3 - c_2b_3) - b_1(d_2c_3 - c_2d_3) + c_1(d_2b_3 - b_2d_3) $$, $$ W_y = a_1(d_2c_3 - c_2d_3) - d_1(a_2c_3 - c_2a_3) + c_1(a_2d_3 - d_2a_3) $$, $$ W_z = a_1(b_2d_3 - d_2b_3) - b_1(a_2d_3 - d_2a_3) + d_1(a_2b_3 - b_2a_3) $$

Equation 1 Coefficients

Equation 2 Coefficients

Equation 3 Coefficients

Solution & Determinants:

Understanding Cramer's Rule

Cramer's Rule is a powerful method in linear algebra used to solve systems of linear equations. It provides a direct way to find the solution in terms of determinants. For a system of equations, Cramer's Rule uses the determinant of the coefficient matrix and determinants of matrices formed by replacing columns with the constant terms.

Imagine you have equations like $ax + by = c$ and $dx + ey = f$. Cramer's Rule helps you find $x$ and $y$ using simple determinant calculations. The denominator is the determinant of the coefficients of $x$ and $y$, while the numerators are determinants where the column for the variable you're solving for is replaced by the constants on the right side of the equations.

This tool is perfect for students and anyone needing to solve linear equations quickly and understand the underlying method. While effective for smaller systems, for very large systems, other methods might be more computationally efficient.

Learn more on Wikipedia.