To analyze the relationship between the number of track meets (x) and the number of races (y) based on the information given, let's address each part of your question in sequence.
Part A: Calculate the slope of the linear equation shown in the graph.
The slope (m) of a line can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
From the points provided:
- Point 1 (x₁, y₁) = (0, 5)
- Point 2 (x₂, y₂) = (1, 7)
Now substituting these points into the slope formula:
\[ m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2 \]
Thus, the slope of the line 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 every additional track meet (increase in x by 1), the number of races (y) increases by 2. This indicates a direct relationship between the number of track meets and the number of races: the more track meets the runner participates in, the more races he competes in, specifically increasing his races by 2 for each meet.
Part C: Interpret the y-intercept in the situation.
The y-intercept occurs where the line crosses the y-axis, which happens when x = 0. From the graph and the points provided, the y-intercept is at (0, 5).
This means that when the runner does not participate in any track meets (0 meets), he has already completed 5 races. This could imply that these races may include training races, previous season races, or races that are not tied to the meets counted in this particular season.
Part D: Write the equation of the line shown on the graph in slope-intercept form.
The slope-intercept form of a linear equation is given by:
\[ y = mx + b \]
Where:
- m = slope
- b = y-intercept
From Part A, we know the slope \(m = 2\), and from Part C, we have the y-intercept \(b = 5\). Substituting these values into the slope-intercept form gives:
\[ y = 2x + 5 \]
Thus, the equation of the line in slope-intercept form is \(y = 2x + 5\).