Bill and Rebecca are playing a game at an arcade. Bill starts the next round of the game with 30 points and earns 4 more points each round. Rebecca starts the next round with 60 points but loses 6 more points each round. Is there a time when they’ll have the same number of points?

Question

The equation below represents the number of points, y, that Rebecca has after x rounds. Use the equation to complete the table with the number of points Rebecca has after 2, 3, and 5 rounds. You can solve on paper and then enter your answer on Zearn.
y=60−6x

Rounds (x) Points (y)
2
3
5
Good work!
Next, graph the values you just found on the coordinate plane to show the relationship between the number of rounds Rebecca plays and the number of points she has.
Rounds (x) Points (y)
2 48
3 42
5 30
Great job!
Next, consider Bill. The equation below represents the number of points, y, that Bill has after x rounds. Use the equation to complete the table with the number of points Bill has after 3, 6, and 9 rounds. You can solve on paper and then enter your answer on Zearn.
y=30+4x

Rounds (x) Points (y)
3
6
9

Answer 1

Let's calculate Rebecca's points using the equation \( y = 60 - 6x \):

For Rebecca:

1. For 2 rounds (x = 2): \[ y = 60 - 6(2) = 60 - 12 = 48 \]

2. For 3 rounds (x = 3): \[ y = 60 - 6(3) = 60 - 18 = 42 \]

3. For 5 rounds (x = 5): \[ y = 60 - 6(5) = 60 - 30 = 30 \]

Table for Rebecca's Points:

| Rounds (x) | Points (y) | |------------|------------| | 2 | 48 | | 3 | 42 | | 5 | 30 |

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Now, let's calculate Bill's points using the equation \( y = 30 + 4x \):

For Bill:

1. For 3 rounds (x = 3): \[ y = 30 + 4(3) = 30 + 12 = 42 \]

2. For 6 rounds (x = 6): \[ y = 30 + 4(6) = 30 + 24 = 54 \]

3. For 9 rounds (x = 9): \[ y = 30 + 4(9) = 30 + 36 = 66 \]

Table for Bill's Points:

| Rounds (x) | Points (y) | |------------|------------| | 3 | 42 | | 6 | 54 | | 9 | 66 |

Summary:

• Rebecca's Points:

  • 2 rounds: 48 points
  • 3 rounds: 42 points
  • 5 rounds: 30 points

• Bill's Points:

  • 3 rounds: 42 points
  • 6 rounds: 54 points
  • 9 rounds: 66 points

Now you can use these values to graph the relationships on a coordinate plane as required!