N-Solve Calculator

What is N-solve?

N-solve (short for "N-variable solve") is a mathematical method for finding solutions to systems of linear equations. These systems can have any number of variables (N) and equations, making it useful for problems in chemistry, physics, engineering, and economics.

The calculator implements the following methods:

• Gaussian elimination for exact solutions
• Matrix row operations to transform the system
• Determinant calculation for consistency checks
• Parametric solutions for underdetermined systems

How to Use the Calculator

1. Enter the number of variables in your system (N)
2. Enter the number of equations in your system (M)
3. For each equation, enter the coefficients for each variable and the constant term
4. Click "Calculate" to solve the system
5. Review the solution and any additional information

The calculator will display:

• The solution set if a unique solution exists
• Parametric solutions if infinitely many solutions exist
• A "No solution" message if the system is inconsistent
• The matrix operations performed during the solution process

Formula Used

The calculator solves the system of linear equations represented by the matrix equation:

Where:

• A is the coefficient matrix (M × N)
• X is the solution vector (N × 1)
• B is the constant vector (M × 1)

The solution is found using Gaussian elimination with partial pivoting:

1. Transform the augmented matrix [A|B] to row-echelon form
2. Check for consistency and determine the solution type
3. Back-substitute to find the solution values

Worked Example

Consider the system of equations:

The calculator would:

1. Create the augmented matrix:
   [2 3 | 8] [4 -1 | 6]
2. Perform row operations to get row-echelon form:
   [1 1.5 | 4] [0 -7 | -10]
3. Back-substitute to find x = 2 and y = 2

The solution is x = 2, y = 2.