Question

melanie and tracy are each finding the equation of the trend line that fits the data in the table below.

participation in the school band
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$$\begin{tabular}{|c|c|} \\hline year & number of students \\\\ \\hline 2010 & 48 \\\\ \\hline 2011 & 52 \\\\ \\hline 2012 & 55 \\\\ \\hline 2013 & 59 \\\\ \\hline \\end{tabular}$$

melanie uses the ordered pairs (2010, 48) and (2013, 59) to find her equation. tracy defines \\(x\\) as the number of years since 2010 and uses the ordered pairs (0, 48) and (3, 59) to find her equation. how will the two girls equations compare?

they will have the same slopes and the same y-intercepts.
they will have the same slopes but different y-intercepts.
they will have different slopes but the same y-intercepts.
they will have different slopes and different y-intercepts.

Explanation:

Calculate the slopes for both equations

$$ m_{\text{Melanie}} = \frac{59 - 48}{2013 - 2010} = \frac{11}{3} $$

$$ m_{\text{Tracy}} = \frac{59 - 48}{3 - 0} = \frac{11}{3} $$

Determine the y-intercepts for both equations

For Melanie, using \(y - y_1 = m(x - x_1)\) with \((2010, 48)\):

$$ y - 48 = \frac{11}{3}(x - 2010) \implies y = \frac{11}{3}x - 7322 $$

For Tracy, using \((0, 48)\) directly gives the y-intercept:

$$ y = \frac{11}{3}x + 48 $$

Compare the slopes and y-intercepts

$$ m_{\text{Melanie}} = m_{\text{Tracy}} = \frac{11}{3} $$

$$ b_{\text{Melanie}} = -7322 eq b_{\text{Tracy}} = 48 $$

Answer:

• They will have the same slopes and the same y-intercepts.
• They will have the same slopes but different y-intercepts. (Correct answer)
• They will have different slopes but the same y-intercepts.
• They will have different slopes and different y-intercepts.