Cramer’s Rule Calculator
Solve systems of linear equations using Cramer’s rule. Enter coefficients and get step-by-step solutions with determinant calculations.
Cramer’s Rule for Linear Systems
Cramer’s rule is a mathematical theorem that provides an explicit solution for systems of linear equations with as many equations as unknowns, provided the system has a unique solution. Named after Swiss mathematician Gabriel Cramer, this method uses determinants to solve linear systems efficiently.
The rule applies when the coefficient matrix is square and has a non-zero determinant. For a system Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constants vector, each variable xi can be calculated as xi = Di / D, where D is the determinant of matrix A, and Di is the determinant of the matrix formed by replacing the i-th column of A with vector b.
This method is particularly useful for small systems (2×2 or 3×3) and provides exact solutions when manual calculation is needed. While computationally intensive for larger systems, Cramer’s rule offers valuable insights into the relationship between determinants and linear system solutions.
How to Use the Cramer’s Rule Calculator
Step 1: Select System Size
Choose whether you want to solve a 2×2 or 3×3 system of linear equations using the dropdown menu. The calculator will automatically adjust the input fields based on your selection.
Step 2: Enter Coefficient Values
Input the coefficients for each variable in your system of equations. Enter the numbers that appear before the variables (x, y, z) in each equation. Use positive or negative decimal numbers as needed.
Step 3: Input Constant Terms
Enter the constant values that appear on the right side of the equals sign for each equation. These represent the results that each linear equation should equal.
Step 4: Calculate and Review Results
Click the calculate button to solve your system using Cramer’s rule. The calculator will display the solution for each variable, show all determinant calculations, and provide a detailed explanation of the solving process.
Step 5: Interpret the Solution
Review the calculated values for each variable and examine the determinant calculations to understand how the solution was derived. If the main determinant equals zero, the system has no unique solution.