Linear System of Equations with Parameters

1. Solving a Linear System of Equations with Parameters by Cramer's Rule

System with Parameters

1. Find the rank of the matrix of coefficients:

Matrix

Matrix

System Solution

System Solution

2. Find the rank of the augmented matrix:

Augmented Matrix

Range of a Matrix

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

Inconsistent System

Consistent Independent System

4. Solve the compatible system by Cramer's rule (also can be solved by the Gauss Elimination method).

Cramer's Rule

Cramer's Rule

System Solution


2. Solving a Linear System of Equations with Parameters by the Gauss Elimination Method

It needs to be determined whether there is any value of m to make the system consistent. If so, solve the system for that value of m.

System of Equations

System Operations

System Operations

System Operations

System Solution

Inconsistent System

Consistent Independent System

System of Equations

System Solution

Examples

System of Equations

System Operations

System Operations

If a = 0.

System Operations

Inconsistent System

If a = −3

System Operations

System Operations

System Solution

Consistent Dependent System

System Operations

System Operations

System Solution

Consistent Independent System

System Operations

System Operations

System Operations

System Solution

System Solution


System of Equations

System Operations

a ≠ 1 r = 3 n = 3

Trivial Solution

System Solution

System Operations

System Operations

System Solutions


System of Equations

System Operations

System Operations

System Solutions

Consistent Independent System

System Operations

System Operations

System Operations

Inconsistent System

System Operations

Consistent Dependent System