Types of Linear Systems

Systems of linear equations can be classified by the number of solutions.

An inconsistent system has no solutions.

  • Its graph is the graph of two parallel lines.
  • In its equations, the slopes are the same and the y-intercepts are different.

y = ½x + 3

y = ½x + 1

2 parallel lines

 

A consistent system has at least one solution. There are 2 types of consistent systems. The first type is an independent system.

An independent system has exactly one solution.

  • Its graph is the graph of 2 lines intersecting in
    one point.
  • In its equations, the slopes are different.

y = ½x + 2

y = x + 3

2 lines with a single point of intersection

 

The second type of a consistent system is a dependent system. A dependent system has infinitely many solutions.

  • Its graph is the graph of one line lying on top of the other - it looks like a single line.
  • In its equations, both the slopes and y-intercepts are the same.

y = ½x + 2

y = x + 6

2 lines directly overlapping each other

 

As a summary:

Inconsistent Consistent and Independent Consistent and Dependent
  • 0 solutions
  • When solved, you will have a single number of x and y.
  • 1 solution
  • When solved, you will have a single number for x and y.
  • Infinite solutions
  • When solved, you will have a statement that is always true, such as 2 = 2.
2 parallel lines 2 lines with a single point of intersection 2 lines directly overlapping each other

There are 3 ways to solve a system of linear equations:

  • Graphing
  • Substitution
  • Elimination

Next, we will solve systems using these methods.

 

 

 

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