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A tub of water is emptied at a rate of 3 gallons per minute. The equation [tex]$y-12=-3(x-1)$[/tex] models the amount of water remaining, where [tex]$x$[/tex] is time (in minutes) and [tex]$y$[/tex] is the amount of water left (in gallons). Analyze the work shown below to determine the initial amount of water.

1. Solve for the [tex]$y$[/tex]-variable.
[tex]\[
\begin{array}{l}
y - 12 = -3(x - 1) \\
y - 12 = -3x + 3 \\
y = -3x + 15
\end{array}
\][/tex]

2. Write the equation using function notation.
[tex]\[
f(x) = -3x + 15
\][/tex]

The tub started with [tex]$\boxed{15}$[/tex] gallons of water.



Answer :

GinnyAnswer
Let's solve the problem step-by-step:

### Step 1: Solve for the [tex]\( y \)[/tex]-variable
The given equation is:
[tex]\[ y - 12 = -3(x - 1) \][/tex]

Our goal is to isolate [tex]\( y \)[/tex]:

1. Distribute [tex]\(-3\)[/tex] on the right-hand side:
[tex]\[ y - 12 = -3x + 3 \][/tex]

2. Add 12 to both sides of the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -3x + 15 \][/tex]

### Step 2: Write the equation in function notation
We can denote the amount of water remaining as a function of time [tex]\( x \)[/tex]:
[tex]\[ f(x) = -3x + 15 \][/tex]

### Step 3: Determine the initial amount of water
To find the initial amount of water, we need to evaluate the function at [tex]\( x = 0 \)[/tex] (which represents the starting point, or initial time):

When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = f(0) = -3(0) + 15 \][/tex]
[tex]\[ y = 15 \][/tex]

Thus, the tub initially had 15 gallons of water.

So, the tub started with 15 gallons of water.