Question
Betsy starts rock climbing at an elevation above sea level. She climbs up at a constant rate per hour. Write an equation to represent this situation. Then use the equation to make a prediction.
This graph represents the linear relationship between the time in hours and Betsy's elevation in meters. For each hour, her elevation increases at a constant rate. After 2 hours, Betsy’s elevation is 630 meters. After 5 hours, her elevation is 1,530 meters. 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 Betsy’s change in elevation for each hour?
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)
y= mx + b
Slope =
Good work!
To represent the relationship between time and Betsy’s elevation, a linear equation can be written in the form y=mx+b. Substitute the slope for m in our equation.
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)
y= mx + b
y=
x+b
Slope = 300
Nice!
Now, solve for b, or the vertical intercept. You can can substitute the x and y-values for a point on the line into the equation to help you. Solve on paper or use the calculator. Then, enter your answer on Zearn.
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)
y=300x+b
b=
This graph represents the linear relationship between the time in hours and Betsy's elevation in meters. For each hour, her elevation increases at a constant rate. After 2 hours, Betsy’s elevation is 630 meters. After 5 hours, her elevation is 1,530 meters. 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 Betsy’s change in elevation for each hour?
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)
y= mx + b
Slope =
Good work!
To represent the relationship between time and Betsy’s elevation, a linear equation can be written in the form y=mx+b. Substitute the slope for m in our equation.
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)
y= mx + b
y=
x+b
Slope = 300
Nice!
Now, solve for b, or the vertical intercept. You can can substitute the x and y-values for a point on the line into the equation to help you. Solve on paper or use the calculator. Then, enter your answer on Zearn.
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)
y=300x+b
b=
Answers
GPT-4o mini
To find the y-intercept \( b \) of the line, we can use one of the given points. Let's use the point \((2, 630)\). We can substitute the values into the equation \( y = mx + b \).
1. We already have the slope \( m = 300 \).
2. Substitute the coordinates of the point into the equation:
\[
630 = 300(2) + b
\]
3. Simplify this:
\[
630 = 600 + b
\]
4. Now, solve for \( b \):
\[
b = 630 - 600
\]
\[
b = 30
\]
Now, we can write the complete equation with the values we have found:
\[
y = 300x + 30
\]
With this equation, you can predict Betsy's elevation at any given time (in hours). For example, if you want to predict her elevation after 4 hours, you would substitute \( x = 4 \):
\[
y = 300(4) + 30 = 1200 + 30 = 1230 \text{ meters}
\]
Thus, her elevation after 4 hours would be 1230 meters.
1. We already have the slope \( m = 300 \).
2. Substitute the coordinates of the point into the equation:
\[
630 = 300(2) + b
\]
3. Simplify this:
\[
630 = 600 + b
\]
4. Now, solve for \( b \):
\[
b = 630 - 600
\]
\[
b = 30
\]
Now, we can write the complete equation with the values we have found:
\[
y = 300x + 30
\]
With this equation, you can predict Betsy's elevation at any given time (in hours). For example, if you want to predict her elevation after 4 hours, you would substitute \( x = 4 \):
\[
y = 300(4) + 30 = 1200 + 30 = 1230 \text{ meters}
\]
Thus, her elevation after 4 hours would be 1230 meters.