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 =
Good work!
Substitute the slope for m in our equation, or the rate of change of the gallons of gas.
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
y=
x+b
Slope = 15
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 =
Good work!
Substitute the slope for m in our equation, or the rate of change of the gallons of gas.
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
y=
x+b
Slope = 15
Answers
GPT-4o mini
Let's break down the problem and fill in the missing parts step by 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.
2. **Finding the Slope (m):**
The slope can be calculated using the formula:
\[
m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's use the points (2, 50) and (4, 80):
- \(y_1 = 50\), \(y_2 = 80\), \(x_1 = 2\), and \(x_2 = 4\)
\[
m = \frac{80 - 50}{4 - 2} = \frac{30}{2} = 15
\]
Therefore, the slope \(m = 15\). This means that for each minute, the gallons of gas in the truck increases by **15 gallons**.
3. **Creating the Equation:**
Using the slope-intercept form \(y = mx + b\):
- We have \(m = 15\).
- Now we need to find \(b\) (the y-intercept). We can substitute one of the points into the equation to solve for \(b\). Using the point (2, 50):
\[
50 = 15(2) + b
\]
\[
50 = 30 + b
\]
\[
b = 50 - 30 = 20
\]
4. **Final Equation:**
Now we can write the equation that represents the situation:
\[
y = 15x + 20
\]
5. **Making a Prediction:**
For example, to predict how many gallons of gas will be in the truck after 6 minutes, we can plug \(x = 6\) into the equation:
\[
y = 15(6) + 20 = 90 + 20 = 110
\]
So after 6 minutes, there will be **110 gallons** of gas in the truck.
In summary:
- The points (2, 50) and (4, 80) provide specific instances of gas in the truck over time.
- The slope of the line is **15 gallons per minute**.
- The equation representing the relationship is \(y = 15x + 20\).
- After 6 minutes, there will be **110 gallons** in the truck.
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.
2. **Finding the Slope (m):**
The slope can be calculated using the formula:
\[
m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's use the points (2, 50) and (4, 80):
- \(y_1 = 50\), \(y_2 = 80\), \(x_1 = 2\), and \(x_2 = 4\)
\[
m = \frac{80 - 50}{4 - 2} = \frac{30}{2} = 15
\]
Therefore, the slope \(m = 15\). This means that for each minute, the gallons of gas in the truck increases by **15 gallons**.
3. **Creating the Equation:**
Using the slope-intercept form \(y = mx + b\):
- We have \(m = 15\).
- Now we need to find \(b\) (the y-intercept). We can substitute one of the points into the equation to solve for \(b\). Using the point (2, 50):
\[
50 = 15(2) + b
\]
\[
50 = 30 + b
\]
\[
b = 50 - 30 = 20
\]
4. **Final Equation:**
Now we can write the equation that represents the situation:
\[
y = 15x + 20
\]
5. **Making a Prediction:**
For example, to predict how many gallons of gas will be in the truck after 6 minutes, we can plug \(x = 6\) into the equation:
\[
y = 15(6) + 20 = 90 + 20 = 110
\]
So after 6 minutes, there will be **110 gallons** of gas in the truck.
In summary:
- The points (2, 50) and (4, 80) provide specific instances of gas in the truck over time.
- The slope of the line is **15 gallons per minute**.
- The equation representing the relationship is \(y = 15x + 20\).
- After 6 minutes, there will be **110 gallons** in the truck.