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Let's solve the problem step by step.

### Problem Breakdown:
- **Rate of draining**: The pump drains 24 gallons per hour.
- **Time given**: After 3 hours, there are 78 gallons left in the tank.

### Part (a): Write an equation to model the water left in the tank, \( y \), after \( x \) hours of draining.

The general equation for the amount of water left after \( x \) hours can be modeled by a linear equation of the form:
\[
y = mx + b
\]
where:
- \( m \) is the rate of change (in this case, the negative rate at which the water is being drained),
- \( b \) is the initial amount of water in the tank.

**Step 1: Determine \( m \) (the slope).**
- The tank drains at a rate of 24 gallons per hour, so \( m = -24 \) (negative because the amount of water is decreasing).

**Step 2: Use the information given to find \( b \).**
- After 3 hours, 78 gallons remain. We can use this information to solve for \( b \) (the initial amount of water).
- Substitute \( x = 3 \), \( y = 78 \), and \( m = -24 \) into the equation:
  \[
  78 = -24(3) + b
  \]
  \[
  78 = -72 + b
  \]
  \[
  b = 150
  \]

So, the equation that models the water left in the tank after \( x \) hours is:
\[
y = -24x + 150
\]

### Part (b): How many gallons were originally in the tank?

The original amount of water in the tank is given by \( b \), which we've calculated as 150 gallons.

### Final Answer:
- **(a)** The equation is \( y = -24x + 150 \).
- **(b)** There were originally 150 gallons in the tank.

Do you have any questions or need further explanation?

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**Relative Questions:**
1. How would the equation change if the pump drained 30 gallons per hour instead of 24?
2. What would the equation be if the tank had 100 gallons left after 5 hours?
3. How do you interpret the slope and y-intercept in this context?
4. If the tank started with 200 gallons, what would the equation be?
5. What happens to the equation if the draining rate slows down over time?

**Tip:** When writing an equation based on a word problem, identifying the rate of change and initial value is key to forming the correct linear model.

Learn how to model the rate of water drainage using a linear equation. Solve the problem step-by-step with a detailed explanation of how to find the initial amount of water in the tank and the rate of drainage. Suitable for students in Grades 7-9.

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