# Types of Solutions to S.O.L.E. Investigations

- Author:
- Shelby Povlick

## Description:

Above you see two linear equations graphed on the same coordinate plane. You have been learning about how to graph systems of equations in order to find their solutions. During the next 3 Investigations, you will discover the different types of solutions there are to systems of linear equations and explore how the differences can be seen between them by looking at the graph and equations.
Both equations are written in slope-intercept form where
The blue equation corresponds with the blue line. This equation will

**not**change during your investigations. The red equation corresponds with the red line. By**manipulating the sliders**(that are also red), you will be able to change the equation and make conjectures about the different types of solutions to systems of linear equations.## Investigation 1: One Solution

1. Drag slider m to change the slope of the red line to different values (except for m=3).
2. Keeping that slider the same, drag the b slider to change the y-intercept of the red line to various values
3. What do you notice about the graphs of the two equations? Do they intersect? If yes, at how many points do they intersect?
4. What do you notice about the equations (blue vs. red)? What is the same or different about the equations (slope, y-intercepts, etc)?
5. Make a conjecture about the equations and graphs of systems of linear equations with

**one solution**.## Investigation 2: No Solutions

1. Drag slider m to change the slope of the red line to m=3.
2. Keeping that slider the same, drag the b slider to change the y-intercept of the red line to various values (except b=2)
3. What do you notice about the graphs of the two equations? Do they intersect? If yes, at how many points do they intersect?
4. What do you notice about the equations (blue vs. red)? What is the same or different about the equations?
5. Make a conjecture about the equations and graphs of systems of linear equations with

**no solutions**.## Investigation 3: Infinitely Many Solutions

1. Drag slider m to change the slope of the red line to m=3.
2. Keeping that slider the same, drag the b slider to change the y-intercept of the red line to b=2.
3. What do you notice about the graphs of the two equations? Do they intersect? If yes, at how many points do they intersect?
4. What do you notice about the equations (blue vs. red)? What is the same or different about the equations?
5. Make a conjecture about the equations and graphs of systems of linear equations with

**infinitely many solutions**.## Question 1:

What type of solution would the following system of equations have? Choose your answer and then on your Investigation Sheet, explain how you know without graphing the system.

## Question 2:

What type of solution would the following system of equations have? Choose your answer and then on your Investigation Sheet, explain how you know without graphing the system.

## Question 3:

What type of solution would the following system of equations have? Choose your answer and then on your Investigation Sheet, explain how you know without graphing the system.