This tool will help you solve systems of linear equations using Cramer’s Rule.

### How to Use the Calculator

Enter the coefficients (a, b, c) and constants (d) for three linear equations of the form ax + by + cz = d. Click the “Calculate” button to solve the system of equations using Cramer’s Rule. The result will display the values of x, y, and z.

### How It Works

Cramer’s Rule solves a system of linear equations using determinants. If you have a system of three equations:

a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3

The calculator computes the determinant of the coefficient matrix (detA) and uses it to find the determinants for matrices with replaced columns. The results are the values of x, y, and z:

detA = a1(b2c3 - c2b3) - b1(a2c3 - c2a3) + c1(a2b3 - b2a3) detX = d1(b2c3 - c2b3) - b1(d2c3 - c2d3) + c1(d2b3 - b2d3) detY = a1(d2c3 - c2d3) - d1(a2c3 - c2a3) + c1(a2d3 - d2a3) detZ = a1(b2d3 - d2b3) - b1(a2d3 - d2a3) + d1(a2b3 - b2a3) x = detX / detA y = detY / detA z = detZ / detA

### Limitations

The method requires that the determinant of the coefficient matrix (detA) is not zero. If detA is zero, the system does not have a unique solution. Additionally, the calculator only works for systems with three equations and three unknowns.

## Use Cases for This Calculator

### Calculate 2×2 system using Cramer’s Rule

Input the coefficients of x and y for both equations, and the calculator will solve the system using Cramer’s Rule, providing you with the values of x and y.

### Calculate 3×3 system using Cramer’s Rule

Enter the coefficients of x, y, and z for the three equations, and the calculator will apply Cramer’s Rule to find the solutions for x, y, and z.

### Verify if system is solvable with Cramer’s Rule

Input the coefficients of the system, and the calculator will check if the determinant of the coefficient matrix is non-zero, indicating solvability through Cramer’s Rule.

### Handle inconsistent systems with Cramer’s Rule

If the system is inconsistent, the calculator will inform you if Cramer’s Rule cannot be applied for such systems, ensuring accurate calculations.

### Detect dependency in the system of equations

The calculator will alert you if the determinant of the coefficient matrix is zero, indicating dependency among the equations and the inability to apply Cramer’s Rule.

### Efficiently solve linear equations with Cramer’s Rule

By inputting the coefficients, you can quickly solve a system of linear equations using Cramer’s Rule, saving you time and effort in manual calculations.

### Understand the step-by-step calculations with Cramer’s Rule

The calculator will provide you with a breakdown of the Cramer’s Rule method, helping you understand the process of solving simultaneous equations in a clear and simplified manner.

### Get accurate solutions for non-linear systems using Cramer’s Rule

Even for non-linear systems, the calculator can accurately apply Cramer’s Rule to determine the values of variables, providing reliable solutions for complex equations.

### Quickly cross-verify results of simultaneous equations with Cramer’s Rule

You can swiftly confirm the solutions obtained through other methods by using Cramer’s Rule, ensuring the correctness of your calculations through a different approach.

### Enhance your mathematical skills with Cramer’s Rule calculations

By using the calculator to practice solving systems of equations with Cramer’s Rule, you can improve your understanding of linear algebra concepts and sharpen your problem-solving abilities.