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Joelle currently has 18 rocks in her collection and gains 4 each week. Lewis currently has 30 rocks in his collection and gains 3 each week. Set up a system of equations to show how many rocks each has in their collection.

Joelle has y = Response area x + Response area

Lewis has y = Response area x + Response area

After how many weeks will Joelle and Lewis have the same number of rocks in their collections? Response area weeks

How many rocks will Joelle and Lewis have when the amount of rocks in their collection is equal? They will each have Response area rocks in their collection.

1 answer

To set up a system of equations for Joelle and Lewis, we can define the variables as follows:

Let \( x \) represent the number of weeks.

For Joelle:

  • Initial rocks: 18
  • Rocks gained per week: 4
  • Equation: \[ y = 4x + 18 \]

For Lewis:

  • Initial rocks: 30
  • Rocks gained per week: 3
  • Equation: \[ y = 3x + 30 \]

Now we have the system of equations:

  1. \( y = 4x + 18 \) (Joelle)
  2. \( y = 3x + 30 \) (Lewis)

To find out after how many weeks Joelle and Lewis will have the same number of rocks, we need to set the two equations equal to each other:

\[ 4x + 18 = 3x + 30 \]

Now, solving for \( x \):

  1. Subtract \( 3x \) from both sides: \[ 4x - 3x + 18 = 30 \] \[ x + 18 = 30 \]

  2. Subtract 18 from both sides: \[ x = 30 - 18 \] \[ x = 12 \]

So, Joelle and Lewis will have the same number of rocks after 12 weeks.

To find out how many rocks they will each have at that time, substitute \( x = 12 \) into either of the original equations. Using Joelle's equation:

\[ y = 4(12) + 18 \] \[ y = 48 + 18 \] \[ y = 66 \]

Thus, they will each have 66 rocks in their collection when the amounts are equal.

Here's a summary:

  • The system of equations is:

    • Joelle: \( y = 4x + 18 \)
    • Lewis: \( y = 3x + 30 \)

  • They will have the same number of rocks after 12 weeks.

  • They will each have 66 rocks at that time.