Solving Systems of Equations

The necessary and sufficient condition for a system of m equations and n unknowns has a solution where the range of the coefficient matrix and the augmented matrix are equal.

  • r = r'     Consistent system.
    • r = r' = n   Consistent independent system.
    • r = r'≠ n   Consistent dependent system.
  • r ≠ r'    Inconsistent system.

Steps to Solve a System of Equations

System of Equations

1. Find the rank of the matrix of coefficients:

Rank of the Matrix

Rank of the Matrix

r(A) = 3

2. Find the rank of the augmented matrix:

Augmented Matrix

Determinant

r(A') = 3

3. Study the obtained information and determine which type of system it is:

System Solution

Consistent Independent System

4. Solve the system if it is not inconsistent, by Cramer's rule or the Gauss elimination method.

Take the system corresponding to the submatrix of order 3, which has a rank of 3 and solve it:

System of Equations

Cramer's Rule

Cramer's Rule

Examples

1.System of Equations

Matrix

Matrix

Rank of Matrix

Rank of Matrix

System of Solution

solución

System of Equations

System Operations

System Operations

System Operations

System Operations


2. System of Equations

Matrix Operations

Rank of Matrix

Matrix Operations

System Solution

Consistent Independent System


3.System of Equations

Study and resolve the system, if possible:

System of Equations

System Operations

System Operations

System Operations

System Solution

Consistent Independent System