Linear Systems: what they are, types and how to solve

Linear systems are sets of equations associated with each other that have the following form:

The left-hand brace is the symbol used to signal that equations are part of a system. The result of the system is given by the result of each equation.

the coefficients a_m, a_m2, a_m3,..., a_n3, a_n2, a_n1 of the unknowns x₁, x_m2,x_m3,..., x_n3, x_n2, x_n1 are real numbers.
At the same time, b is also a real number which is called an independent term.

Homogeneous linear systems are those whose independent term is equal to 0 (zero): a₁x₁ + the₂x₂ = 0.
Therefore, those with an independent term different from 0 (zero) indicate that the system is not homogeneous: a₁x₁ + the₂x₂ = 3.

Classification

Linear systems can be classified according to the number of possible solutions. Remembering that the solution of the equations is found by replacing the variables with values.

• Possible and Determined System (SPD): there is only one possible solution, which happens when the determinant is non-zero (D ≠ 0).
• Possible and Indeterminate System (SPI): the possible solutions are endless.
• Impossible System (SI): it is not possible to present any kind of solution.

At matrices associated with a linear system can be complete or incomplete. The matrices that consider the independent terms of the equations are complete.

Linear systems are classified as normal when the number of equations is the same as the number of unknowns. Also, when the determinant of the incomplete matrix of that system is not equal to zero.

Solved Exercises

Let's solve each equation step by step in order to classify them into SPD, SPI or SI.

Example 1 - Linear System with 2 Equations

Example 2 - Linear System with 3 Equations

If D = 0, we can be facing an SPI or an SI.

Read:

• Equation Systems
• 1st degree Equation Systems - Exercises
• Determinants
• First Degree Equation
• Second degree equation
• Competing Lines