Sure, let's break it down step by step to determine the initial amount of water in the tub.
1. Solve for the \( y \)-variable:
Given the equation \( y - 12 = -3(x - 1) \), we need to isolate \( y \).
[tex]\[
y - 12 = -3(x - 1)
\][/tex]
Distribute the \(-3\) across the terms inside the parenthesis:
[tex]\[
y - 12 = -3x + 3
\][/tex]
Then, add 12 to both sides of the equation to solve for \( y \):
[tex]\[
y - 12 + 12 = -3x + 3 + 12
\][/tex]
Simplifying this gives:
[tex]\[
y = -3x + 15
\][/tex]
2. Write the equation using function notation:
The equation \( y = -3x + 15 \) can be written using function notation as \( f(x) = -3x + 15 \).
3. Determine the initial amount of water:
To find out the initial amount of water in the tub, we need to evaluate \( f(x) \) at \( x = 0 \) (when time \( t = 0 \) minutes).
[tex]\[
f(0) = -3(0) + 15
\][/tex]
Simplifying this, we get:
[tex]\[
f(0) = 15
\][/tex]
So, the tub started with [tex]\( \boxed{15} \)[/tex] gallons of water.
