To solve this problem, let us analyze the given situation:

Problem:

Muriel collects rainwater in a barrel:

• The barrel itself weighs 20 pounds.
• When there are 10 gallons of water, the total weight is 103.4 pounds.
• When there are 20 gallons of water, the total weight is 186.8 pounds.

We need to find the equation representing this situation, which will be a linear equation of the form: $y = mx + b$ where:

• $x$ is the number of gallons of water,
• $y$ is the total weight,
• $m$ is the rate of change (weight per gallon),
• $b$ is the y-intercept (the barrel's weight without any water).

Step 1: Find the slope $m$ (weight per gallon of water):

The slope is given by the rate of change between the points: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the given points $(10, 103.4)$ and $(20, 186.8)$: $m = \frac{186.8 - 103.4}{20 - 10} = \frac{83.4}{10} = 8.34 \, \text{pounds per gallon}.$

Step 2: Find the y-intercept $b$:

The y-intercept represents the total weight when there is no water ($x = 0$). We know: $y = mx + b$ Using the point $(10, 103.4)$ and $m = 8.34$: $103.4 = 8.34(10) + b$ $103.4 = 83.4 + b$ $b = 103.4 - 83.4 = 20.$

Step 3: Write the equation:

The equation is: $y = 8.34x + 20$

Final Answer:

The equation that models this situation is: $y = 8.34x + 20$

Would you like me to explain how this equation matches the graph or clarify anything further?

Follow-up Questions:

1. How do we interpret the slope ($8.34$) in this context?
2. What does the y-intercept ($20$) represent in this problem?
3. If there were 15 gallons of water, what would be the total weight?
4. How would the equation change if the barrel's weight was 25 pounds instead of 20?
5. What is the relationship between this equation and its graph?

Tip:

In word problems involving linear equations, always identify the variables and check if they relate linearly. Start with finding the slope and y-intercept to write the equation.