All, Some, or No Solutions

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Investigation 1: Infinitely Many Solutions.

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

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Investigation 2: One Solution.

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

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Investigation 3: No Solutions.

  • Drag slider m to change the slope of the red line to m=3. ​
  •  Keeping that slider the same, drag the b slider to change the y-intercept of the red line to various values (except b=2) ​
  •  What do you notice about the graphs of the two equations? Do they intersect? If yes, at how many points do they intersect? ​
  •  What do you notice about the equations (blue vs. red)? What is the same or different about the equations? ​
  •  Make a conjecture about the equations and graphs of systems of linear equations with no solutions.

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What type of solution would the following system of equations have?

Explain how you know this is true.

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Explain how you know this is true.

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Explain how you know this is true.