A rectangular bird feeder costs $18.00. A cylindrical bird feeder costs $24.00. The expected cost to keep the rectangular bird feeder filled is $3.00 per week. The expected cost the keep the cylindrical bird feeder filled is $2.00 per week. The equation below models the break-even point.

18 + 3*x* = 24 + 2*x* 

What does *x* represent?

the total cost to fill the rectangular bird feeder after any number of weeks

the total cost to fill the cylindrical bird feeder after any number of weeks

the number of weeks the bird feeders are filled

the number of bird feeders purchased each week

The given equation models the break-even point:

\[ 18 + 3x = 24 + 2x \]

To understand what $ x $ represents, let's analyze the components of the equation:

- $ 18 $: Initial cost of the rectangular bird feeder.
- $ 3x $: Cost to keep the rectangular bird feeder filled over $ x $ weeks (since it costs $3.00 per week).
- $ 24 $: Initial cost of the cylindrical bird feeder.
- $ 2x $: Cost to keep the cylindrical bird feeder filled over $ x $ weeks (since it costs $2.00 per week).

The equation balances the total cost for both bird feeders over time. 

Solving for $ x $:

\[ 18 + 3x = 24 + 2x \]
\[ 3x - 2x = 24 - 18 \]
\[ x = 6 \]

So, $ x = 6 $ represents the point in time (in weeks) where the total cost of the rectangular bird feeder equals the total cost of the cylindrical bird feeder. Hence, $ x $ represents:

**the number of weeks the bird feeders are filled**

Would you like further details or have any questions?

Here are 5 questions you might be interested in:

1. How do you solve linear equations with variables on both sides?
2. What is the significance of the break-even point in cost analysis?
3. How can you model real-life problems using linear equations?
4. What are some applications of break-even analysis in business?
5. How do different costs (fixed vs. variable) impact the total cost over time?

**Tip:** When dealing with equations that model real-life situations, clearly identify what each term represents to understand the meaning of the variables and the overall equation.

Explore the break-even point in cost analysis through a detailed example. Learn how to solve a linear equation modeling the costs of rectangular and cylindrical bird feeders over time. Suitable for Grades 7-9.

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