Question
A track runner counted the number of races he had during a season. The graph shows the relationship between the number of track meets, x, and the number of races, y.
coordinate plane with the x axis labeled number of track meets and the y axis labeled number of races with a line that passes through the points 0 comma 5 and 1 comma 7
Part A: Calculate the slope of the linear equation shown in the graph. Show all necessary work. (3 points)
Part B: What does the slope mean for the relationship between the number of races and the number of track meets? (3 points)
Part C: Interpret the y-intercept in the situation. (3 points)
Part D: Write the equation of the line shown on the graph in slope-intercept form. (3 points)
coordinate plane with the x axis labeled number of track meets and the y axis labeled number of races with a line that passes through the points 0 comma 5 and 1 comma 7
Part A: Calculate the slope of the linear equation shown in the graph. Show all necessary work. (3 points)
Part B: What does the slope mean for the relationship between the number of races and the number of track meets? (3 points)
Part C: Interpret the y-intercept in the situation. (3 points)
Part D: Write the equation of the line shown on the graph in slope-intercept form. (3 points)
Answers
GPT-4o mini
To solve the problem, let's address each part one by one.
### Part A: Calculate the slope of the linear equation.
The slope \( m \) of a line is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Given the two points from the graph:
- Point 1: \( (0, 5) \) (let's denote this as \( (x_1, y_1) \))
- Point 2: \( (1, 7) \) (let's denote this as \( (x_2, y_2) \))
Substituting the coordinates of the points into the slope formula:
\[
m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2
\]
Therefore, the slope of the linear equation is \( 2 \).
### Part B: What does the slope mean for the relationship between the number of races and the number of track meets?
The slope of \( 2 \) means that for each additional track meet the runner participates in, the number of races he runs increases by \( 2 \). In other words, there is a direct relationship between track meets and races; more meets lead to more races.
### Part C: Interpret the y-intercept in the situation.
The y-intercept is the value of \( y \) when \( x = 0 \). From the graph, we can see that when \( x = 0 \), \( y = 5 \).
This means that even when the runner has not participated in any track meets, he still has run \( 5 \) races. This could imply that these races could be from prior seasons, training, or other events outside of the counted meets.
### Part D: Write the equation of the line shown on the graph in slope-intercept form.
The slope-intercept form of a line is given by:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept. We already calculated the slope (\( m = 2 \)) and identified the y-intercept (\( b = 5 \)).
Substituting these values into the equation:
\[
y = 2x + 5
\]
Thus, the equation of the line in slope-intercept form is \( y = 2x + 5 \).
### Part A: Calculate the slope of the linear equation.
The slope \( m \) of a line is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Given the two points from the graph:
- Point 1: \( (0, 5) \) (let's denote this as \( (x_1, y_1) \))
- Point 2: \( (1, 7) \) (let's denote this as \( (x_2, y_2) \))
Substituting the coordinates of the points into the slope formula:
\[
m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2
\]
Therefore, the slope of the linear equation is \( 2 \).
### Part B: What does the slope mean for the relationship between the number of races and the number of track meets?
The slope of \( 2 \) means that for each additional track meet the runner participates in, the number of races he runs increases by \( 2 \). In other words, there is a direct relationship between track meets and races; more meets lead to more races.
### Part C: Interpret the y-intercept in the situation.
The y-intercept is the value of \( y \) when \( x = 0 \). From the graph, we can see that when \( x = 0 \), \( y = 5 \).
This means that even when the runner has not participated in any track meets, he still has run \( 5 \) races. This could imply that these races could be from prior seasons, training, or other events outside of the counted meets.
### Part D: Write the equation of the line shown on the graph in slope-intercept form.
The slope-intercept form of a line is given by:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept. We already calculated the slope (\( m = 2 \)) and identified the y-intercept (\( b = 5 \)).
Substituting these values into the equation:
\[
y = 2x + 5
\]
Thus, the equation of the line in slope-intercept form is \( y = 2x + 5 \).