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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.

a) Joelle has y = ______x + _________

7 answers

To set up a system of equations for Joelle and Lewis based on the information provided, we first need to define the variables:

  • Let \( x \) represent the number of weeks that pass.
  • Let \( y \) represent the number of rocks in their collection.

For Joelle:

  • Joelle currently has 18 rocks and gains 4 rocks each week.
  • Therefore, the equation for Joelle can be represented as:

\[ y = 4x + 18 \]

For Lewis:

  • Lewis currently has 30 rocks and gains 3 rocks each week.
  • Therefore, the equation for Lewis can be represented as:

\[ y = 3x + 30 \]

System of Equations:

Thus, the system of equations is:

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

To answer part (a), we fill in Joelle's equation:

a) Joelle has \( y = 4x + 18 \).

Lewis has y = _______x + _______

For Lewis, we established that he currently has 30 rocks and gains 3 rocks each week. Therefore, the equation for Lewis can be represented as:

\[ y = 3x + 30 \]

To fill in your prompt, we express it as:

Lewis has \( y = 3x + 30 \).

So, the answer is:

Lewis has \( y = 3x + 30 \).

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

To find out when Joelle and Lewis will have the same number of rocks in their collections, we can set their equations equal to each other:

From the earlier equations:

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

Setting them equal:

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

Now, we will solve for \( x \):

  1. Subtract \( 3x \) from both sides:

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

Which simplifies to:

\[ x + 18 = 30 \]

  1. Next, subtract 18 from both sides:

\[ x = 30 - 18 \]

This results in:

\[ x = 12 \]

Conclusion:

Joelle and Lewis will have the same number of rocks in 12 weeks.

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

To find out how many rocks Joelle and Lewis will have when they have the same number of rocks, we can substitute the value of \( x = 12 \) (the number of weeks until they have the same amount) back into either one of their equations.

Let's use Joelle's equation:

\[ y = 4x + 18 \]

Substituting \( x = 12 \):

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

Now, let's confirm this by substituting \( x = 12 \) into Lewis's equation:

\[ y = 3x + 30 \]

Substituting \( x = 12 \):

\[ y = 3(12) + 30 \] \[ y = 36 + 30 \] \[ y = 66 \]

Conclusion:

When Joelle and Lewis have the same number of rocks, they will each have 66 rocks in their collection.