Let's break this down step-by-step.

Chores and their Values:

• Washing dishes, vacuuming, doing the laundry, and taking out the trash = $1 each
• Cleaning the 4 toilets and sinks = $2 each
• Bathtubs = $3 each (since you get an extra $1 for bathtubs)

Define the number of times you do each task:

Let:

• $x_1$ be the number of times you do the washing dishes, vacuuming, laundry, and trash chores ($1 each).
• $x_2$ be the number of times you do the toilets and sinks chores ($2 each).
• $x_3$ be the number of times you clean the bathtubs ($3 each).

We need to find the values of $x_1$, $x_2$, and $x_3$ that will total $145.

Total earnings equation:

$x_1 + 2x_2 + 3x_3 = 145$

Now, let's explore three possible solutions by setting different values for two variables and solving for the third.

Answer 1: Let's say you clean the bathtubs 10 times:

30 + x_1 + 2x_2 = 145 \\ x_1 + 2x_2 = 115$$ Now, let’s test values for $$x_2$$: - If $$x_2 = 50$$: $$x_1 + 2(50) = 115 \quad \Rightarrow \quad x_1 = 15$$ So, the answer is $$x_1 = 15$$, $$x_2 = 50$$, $$x_3 = 10$$. ### Answer 2: Let's say you clean the bathtubs 15 times: $$3(15) + x_1 + 2x_2 = 145 \\ 45 + x_1 + 2x_2 = 145 \\ x_1 + 2x_2 = 100$$ - If $$x_2 = 40$$: $$x_1 + 2(40) = 100 \quad \Rightarrow \quad x_1 = 20$$ So, the answer is $$x_1 = 20$$, $$x_2 = 40$$, $$x_3 = 15$$. ### Answer 3: Let's say you clean the bathtubs 20 times: $$3(20) + x_1 + 2x_2 = 145 \\ 60 + x_1 + 2x_2 = 145 \\ x_1 + 2x_2 = 85$$ - If $$x_2 = 35$$: $$x_1 + 2(35) = 85 \quad \Rightarrow \quad x_1 = 15$$ So, the answer is $$x_1 = 15$$, $$x_2 = 35$$, $$x_3 = 20$$. ### Summary of the 3 answers: 1. $$x_1 = 15$$, $$x_2 = 50$$, $$x_3 = 10$$ 2. $$x_1 = 20$$, $$x_2 = 40$$, $$x_3 = 15$$ 3. $$x_1 = 15$$, $$x_2 = 35$$, $$x_3 = 20$$ Would you like more details or clarification on any of these steps? --- ### Follow-up questions: 1. How does changing the number of bathtubs cleaned affect the solution? 2. What would happen if you did more of the $1 chores and fewer of the $2 or $3 chores? 3. How can you represent this problem graphically? 4. What happens if you add an extra $1 for washing the dishes? 5. How could you optimize your chores to reduce the total number of times you need to do them? **Tip**: You can try testing different values for one variable and solving for the others, as we've done here. This helps you see a range of possible solutions quickly!