Question: A tub of water is emptied at a rate of 3 gallons per minute. The equation $y - 12 = -3(x - 1)$ models the amount of water remaining, where $x$ is time (in minutes) and $y$ is the amount of water left (in gallons). Analyze the work shown below to determine the initial amount of water.
1. Solve for the $y$-variable.
$y - 12 = -3(x - 1)$
$y - 12 = -3x + 3$
$y = -3x + 15$
2. Write the equation using function notation.
$f(x) = -3x + 15$
The tub started with [ ] gallons of water.
1. **State the problem:** We want to find the initial amount of water in the tub, which means the amount of water at time $x = 0$ minutes.
2. **Given equation:** The amount of water remaining is modeled by
$$y - 12 = -3(x - 1)$$
3. **Solve for $y$:**
$$y - 12 = -3x + 3$$
$$y = -3x + 3 + 12$$
$$y = -3x + 15$$
4. **Function notation:** We can write this as
$$f(x) = -3x + 15$$
5. **Find initial amount:** The initial amount of water is the amount when $x = 0$:
$$f(0) = -3(0) + 15 = 15$$
6. **Interpretation:** The tub started with **15 gallons** of water.
**Final answer:** The tub started with 15 gallons of water.