To find the equation of the line passing through the points \((2, 5)\) and \((-6, -6)\), we can use the two-point form of the line equation. First, we need to determine the slope \(m\) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, the points are \((x_1, y_1) = (2, 5)\) and \((x_2, y_2) = (-6, -6)\).
Substituting the values into the formula:
\[ m = \frac{-6 - 5}{-6 - 2} = \frac{-11}{-8} = \frac{11}{8} \]
Now that we have the slope, we can use the point-slope form of the line equation, which is:
\[ y - y_1 = m(x - x_1) \]
Using the point \((2, 5)\) and the slope \(\frac{11}{8}\):
\[ y - 5 = \frac{11}{8}(x - 2) \]
Distributing the slope:
\[ y - 5 = \frac{11}{8}x - \frac{22}{8} \]
Simplifying \(-\frac{22}{8}\) to \(-\frac{11}{4}\):
\[ y - 5 = \frac{11}{8}x - \frac{11}{4} \]
Adding 5 (which is \( \frac{20}{4} \)) to both sides:
\[ y = \frac{11}{8}x - \frac{11}{4} + \frac{20}{4} = \frac{11}{8}x + \frac{9}{4} \]
Thus, the equation of the line is:
\[ y = \frac{11}{8}x + \frac{9}{4} \]
Looking at the multiple-choice answers you've provided, this corresponds to:
𝑦 = 11/8 𝑥 + 9/4
So the correct response is \(y = \frac{11}{8}x + \frac{9}{4}\).
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