To analyze the equations that Melanie and Tracy will derive, we'll start by understanding how both represent their data and their axes.
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Melanie's approach:
- She uses the ordered pairs (2010, 48) and (2013, 59).
- To find the slope (m), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{59 - 48}{2013 - 2010} = \frac{11}{3} \]
- Using the point-slope form of the equation, we can find her line: \[ y - 48 = \frac{11}{3}(x - 2010) \]
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Tracy's approach:
- She defines \( x \) as the number of years since 2010, using the ordered pairs (0, 48) and (3, 59).
- Again, we compute the slope: \[ m = \frac{59 - 48}{3 - 0} = \frac{11}{3} \]
- The line equation becomes: \[ y - 48 = \frac{11}{3}(x - 0) \]
From these calculations, we see that both Melanie and Tracy compute the same slope of \(\frac{11}{3}\). The y-intercept, however, differs due to their use of different \( x \) values.
- Melanie's y-intercept can be derived from her original equation, where the y-intercept can be calculated as seen in her equation with respect to the year.
- Tracy's y-intercept will allow her to express the trend relative to her transformed \( x \).
Since both girls have the same slope, while their y-intercepts will be different (as they define \( x \) differently regarding the starting year), we can conclude that:
The correct answer is: They will have the same slopes but different y-intercepts.