B. The Number of Solutions to a System of Linear Equations

B. The Number of Solutions to a System of Linear Equations

As you saw when graphing linear systems in Lesson 8.1, not every system will have exactly one solution. The following investigation explores this idea.

Investigation

The equations of three lines are shown.

1. Determine the slope and y-intercept of each line.
   
   

   1. What is the relationship between lines A and B?
      
      

   2. What is the relationship between lines A and C?
      
      

2. Use substitution or elimination to solve the system formed by lines A and B.
   
   

   1. Explain how the result is different from other solutions you've seen.
      
      

   2. Use the information from 1a. to explain the result.
      
      

3. Use substitution or elimination to solve the system formed by lines A and C.
   
   

   1. Explain how the result is different from other solutions you've seen.
      
      

   2. Use the information from 1b. to explain the result.
      
      

4. Suggest a general strategy for determining when a system of equations will have
   
   

   1. no solution
      
      

   2. an infinite number of solutions