Types of Linear Systems

Background In slope-intercept form, the equation of a line is given by the form: y = mx + b m represents what? b represents what?

Introduction A system of linear equations consists of two or more linear equations graphed on the same coordinate plane. In this presentation, we will be looking at systems with two linear equations.

Introduction All three types of systems of linear equations we will discuss can be seen in this picture: There are three types of systems of linear equations: Systems with no solution Systems with exactly one solution Systems with infinitely many solutions

System with No Solution In the picture, the highlighted rails are like lines that do not intersect as they are parallel. A system with no solution is a system in which the graphs do not cross or touch anywhere. In other words, the graphs do not intersect; they are parallel. A system with no solution is sometimes referred to as an inconsistent system.

System with No Solution The picture to the right is an example of a system with no solution.

System with Exactly One Solution In the picture, the highlighted rails are like lines that cross at exactly one place. A system with exactly one solution is a system in which the graphs cross or touch at exactly one place. A system with exactly one solution is sometimes referred to as an independent system.

System with Exactly One Solution The picture to the right is an example of a system with exactly one solution.

System with Infinitely Many Solutions In the picture, the highlighted rail looks like one line. The graph of a system with infinitely many solutions will look like one line when graphed. A system with infinitely many solutions consists of the same line twice. In other words, the lines overlap everywhere. A system with infinitely many solutions is sometimes referred to as a dependent system.

System with Infinitely Many Solutions The picture to the right is an example of a system with infinitely many solutions.

Enrichment Do you see examples of systems of equations in the picture to the right?

Enrichment Do you see examples of systems of equations in the picture to the right?

Enrichment Do you see examples of systems of equations in the picture to the right?

Summary What does it mean if a system of linear equations has no solution? What does it mean if a system of linear equations has exactly one solution? What does it mean if a system of linear equations has infinitely many solutions?