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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] 15 [/tex] gallons of water.

Sagot :

GinnyAnswer
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.
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