Question
Destiny fills her truck with gas. The truck has some gas in it, and Destiny fills it at a constant rate each minute. Write an equation that represents this situation. Then use the equation to make a prediction.
This graph represents the linear relationship between the time in minutes and the gallons of gas in the truck. For each minute, the gallons of gas increases at a constant rate. What do the points (2, 50) and (4, 80) represent?
x
y
(2, 50)(4, 80)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Time (minutes)
Gallons of gas
The point (2, 50) means that after
minutes, there are
gallons of gas in the truck.
The point (4, 80) means that after
minutes, there are
gallons of gas in the truck.
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?
x
y
(2, 50)(4, 80)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Time (minutes)
Gallons of gas
y= mx + b
Slope =
This graph represents the linear relationship between the time in minutes and the gallons of gas in the truck. For each minute, the gallons of gas increases at a constant rate. What do the points (2, 50) and (4, 80) represent?
x
y
(2, 50)(4, 80)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Time (minutes)
Gallons of gas
The point (2, 50) means that after
minutes, there are
gallons of gas in the truck.
The point (4, 80) means that after
minutes, there are
gallons of gas in the truck.
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?
x
y
(2, 50)(4, 80)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Time (minutes)
Gallons of gas
y= mx + b
Slope =
Answers
GPT-4o mini
To analyze the situation, we can start with the given points (2, 50) and (4, 80).
### Step 1: Understanding the Points
The point (2, 50) indicates that after **2 minutes**, there are **50 gallons** of gas in the truck.
The point (4, 80) indicates that after **4 minutes**, there are **80 gallons** of gas in the truck.
### Step 2: Finding the Slope (m)
The slope \( m \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (2, 50) and (4, 80):
- \( y_2 = 80 \)
- \( y_1 = 50 \)
- \( x_2 = 4 \)
- \( x_1 = 2 \)
Plugging in these values:
\[
m = \frac{80 - 50}{4 - 2} = \frac{30}{2} = 15
\]
So, the slope \( m \) is **15 gallons per minute**.
### Step 3: Writing the Equation
We can write the equation of the line in the slope-intercept form \( y = mx + b \). We have \( m = 15 \), but we also need to find \( b \) (the y-intercept).
To find \( b \), we can use one of the points. Let's use (2, 50):
\[
y = mx + b
\]
\[
50 = 15(2) + b
\]
\[
50 = 30 + b
\]
\[
b = 50 - 30 = 20
\]
### Final Equation
Now we can write the equation:
\[
y = 15x + 20
\]
### Prediction
Using this equation, if we want to predict how many gallons of gas will be in the truck after a certain number of minutes, we can simply substitute \( x \) with the number of minutes. For example:
If \( x = 6 \):
\[
y = 15(6) + 20 = 90 + 20 = 110
\]
This means that after **6 minutes**, there will be **110 gallons** of gas in the truck.
### Summary
- The slope **(m)** is **15 gallons per minute**.
- The equation representing the situation is \( y = 15x + 20 \).
### Step 1: Understanding the Points
The point (2, 50) indicates that after **2 minutes**, there are **50 gallons** of gas in the truck.
The point (4, 80) indicates that after **4 minutes**, there are **80 gallons** of gas in the truck.
### Step 2: Finding the Slope (m)
The slope \( m \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (2, 50) and (4, 80):
- \( y_2 = 80 \)
- \( y_1 = 50 \)
- \( x_2 = 4 \)
- \( x_1 = 2 \)
Plugging in these values:
\[
m = \frac{80 - 50}{4 - 2} = \frac{30}{2} = 15
\]
So, the slope \( m \) is **15 gallons per minute**.
### Step 3: Writing the Equation
We can write the equation of the line in the slope-intercept form \( y = mx + b \). We have \( m = 15 \), but we also need to find \( b \) (the y-intercept).
To find \( b \), we can use one of the points. Let's use (2, 50):
\[
y = mx + b
\]
\[
50 = 15(2) + b
\]
\[
50 = 30 + b
\]
\[
b = 50 - 30 = 20
\]
### Final Equation
Now we can write the equation:
\[
y = 15x + 20
\]
### Prediction
Using this equation, if we want to predict how many gallons of gas will be in the truck after a certain number of minutes, we can simply substitute \( x \) with the number of minutes. For example:
If \( x = 6 \):
\[
y = 15(6) + 20 = 90 + 20 = 110
\]
This means that after **6 minutes**, there will be **110 gallons** of gas in the truck.
### Summary
- The slope **(m)** is **15 gallons per minute**.
- The equation representing the situation is \( y = 15x + 20 \).