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 answer the questions regarding the relationship between the number of track meets and the number of races, let's analyze the points given in the graph.
### Part A: Calculate the slope of the linear equation shown in the graph.
The coordinates provided are:
- Point 1: (0, 5)
- Point 2: (1, 7)
The slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting our points into the formula:
\[
m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2
\]
So, 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 (each increase of 1 in \(x\)), the number of races (the value of \(y\)) increases by 2. In practical terms, this indicates that the runner participates in 2 additional races for each track meet he attends.
### Part C: Interpret the y-intercept in the situation.
The y-intercept is the value of \(y\) when \(x = 0\). From the point (0, 5), we can see that when the runner does not attend any track meets, he has participated in 5 races. This could imply that the runner either has pre-existing commitments (e.g., races that he participates in outside of the track meets) or is possibly including early season races that take place before the meets start.
### 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 the equation:
\[
y = mx + b
\]
Where \(m\) is the slope and \(b\) is the y-intercept. From our previous calculations, we have:
- Slope \(m = 2\)
- Y-intercept \(b = 5\)
Substituting these values into the equation, we get:
\[
y = 2x + 5
\]
Therefore, the equation of the line shown on the graph in slope-intercept form is \(y = 2x + 5\).
### Part A: Calculate the slope of the linear equation shown in the graph.
The coordinates provided are:
- Point 1: (0, 5)
- Point 2: (1, 7)
The slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting our points into the formula:
\[
m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2
\]
So, 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 (each increase of 1 in \(x\)), the number of races (the value of \(y\)) increases by 2. In practical terms, this indicates that the runner participates in 2 additional races for each track meet he attends.
### Part C: Interpret the y-intercept in the situation.
The y-intercept is the value of \(y\) when \(x = 0\). From the point (0, 5), we can see that when the runner does not attend any track meets, he has participated in 5 races. This could imply that the runner either has pre-existing commitments (e.g., races that he participates in outside of the track meets) or is possibly including early season races that take place before the meets start.
### 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 the equation:
\[
y = mx + b
\]
Where \(m\) is the slope and \(b\) is the y-intercept. From our previous calculations, we have:
- Slope \(m = 2\)
- Y-intercept \(b = 5\)
Substituting these values into the equation, we get:
\[
y = 2x + 5
\]
Therefore, the equation of the line shown on the graph in slope-intercept form is \(y = 2x + 5\).