Section C Section C Checkpoint

Problem 1

{y=3x+5y=3(x+1)\begin{cases} y = 3x + 5 \\ y = 3(x + 1) \end{cases}

  1. How many solutions does this system have? Explain your reasoning without solving the system.
  2. Based on the number of solutions, describe the graph of this system.
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Solution
  1. No solutions. Sample reasoning: The second equation is equivalent to y=3x+3y = 3x + 3. This shows that the 2 equations have the same slope and different yy-intercepts, so there is no solution.
  2. The graphs of the lines are parallel.
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Sample Response
  1. No solutions. Sample reasoning: The second equation is equivalent to y=3x+3y = 3x + 3. This shows that the 2 equations have the same slope and different yy-intercepts, so there is no solution.
  2. The graphs of the lines are parallel.

Problem 2

  1. In a card game, each round you earn either 3 points or 5 points depending on the cards you play. After 5 rounds you have 19 points.

    Use xx for the number of 3 point rounds and yy for the number of 5 point rounds. Write a system of 2 equations that describes this situation.

  2. Another system is solved by the point (7,10)(7,10). Explain how you can check that this solution is correct.
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Solution
  1. {3x+5y=19x+y=5\begin{cases} 3x + 5y &= 19\\ x + y &= 5 \end{cases}(or equivalent)
  2. Sample response: The values make both equations true. Substitute 7 for xx and 10 for yy in the original equations and check that each side of the equations are equal to the other side.
Show Sample Response
Sample Response
  1. {3x+5y=19x+y=5\begin{cases} 3x + 5y &= 19\\ x + y &= 5 \end{cases}(or equivalent)
  2. Sample response: The values make both equations true. Substitute 7 for xx and 10 for yy in the original equations and check that each side of the equations are equal to the other side.