Leo fills his boat with gas. The boat has some gas in it, and Leo fills it at a constant rate each minute. Write an equation that represents this situation. Then use the equation to make a prediction.

Question

This graph represents the linear relationship between the time in minutes and the gallons of gas in the boat. For each minute, the gallons of gas increases at a constant rate. What do the points (4, 50) and (10, 80) represent?
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(4, 50)(10, 80)
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Time (minutes)
Gallons of gas
The point (4, 50) means that after
minutes, there are
gallons of gas in the boat.

The point (10, 80) means that after
minutes, there are
gallons of gas in the boat.
Great job!
Let’s write an equation to represent this line in the form y=mx+b. Start by finding m, or the slope. What is the slope of this line? In other words, what is the change in gallons of gas for each minute?
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(4, 50)(10, 80)
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y= mx + b

Slope =

Answer 1

To write an equation representing the linear relationship between the time (in minutes) and the gallons of gas in the boat, we can start by identifying the slope \(m\). The slope represents the change in gallons of gas for each minute.

To find the slope \(m\), use the two points given: \((4, 50)\) and \((10, 80)\).

The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For the points:

• \((x_1, y_1) = (4, 50)\)
• \((x_2, y_2) = (10, 80)\)

Substituting these values into the slope formula:

\[ m = \frac{80 - 50}{10 - 4} = \frac{30}{6} = 5 \]

So, the slope \(m\) is 5. This means the gallons of gas increases by 5 gallons for each minute.

Next, we can write the equation in the form \(y = mx + b\). We now know \(m\) is 5. We can use one of the points to find \(b\), the y-intercept. Let's use the point \((4, 50)\).

Using the equation:

\[ y = mx + b \]

Substituting the known values:

\[ 50 = 5(4) + b \]

Calculating further:

\[ 50 = 20 + b \] \[ b = 50 - 20 \] \[ b = 30 \]

Now we have both \(m\) and \(b\). The equation representing the relationship is:

\[ y = 5x + 30 \]

Predicting Gallons of Gas

Now you can use this equation to predict the gallons of gas at any time \(x\).

For example, if you want to find out how much gas there will be in the boat after 15 minutes, substitute \(x = 15\):

\[ y = 5(15) + 30 = 75 + 30 = 105 \]

Thus, after 15 minutes, there will be 105 gallons of gas in the boat.

Summary of Points

The point \((4, 50)\) means that after 4 minutes, there are 50 gallons of gas in the boat.

The point \((10, 80)\) means that after 10 minutes, there are 80 gallons of gas in the boat.