To find the equation of the line that is perpendicular to the given line and passes through the point (4, 11), we first need to identify the slope of the given line.
The given equation is:
\[ Y = \frac{5}{8}x + 7 \]
From this equation, we can see that the slope (m) of the line is \( \frac{5}{8} \).
Step 1: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of \( \frac{5}{8} \) is:
\[ m = -\frac{8}{5} \]
Step 2: Use the point-slope form to determine the equation of the new line
We will use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point the line passes through, which is (4, 11), and \( m \) is the slope we just calculated. Plugging in the values:
\[ y - 11 = -\frac{8}{5}(x - 4) \]
Step 3: Rearranging to slope-intercept form
Now, we will simplify this equation:
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Distribute \(-\frac{8}{5}\): \[ y - 11 = -\frac{8}{5}x + \frac{32}{5} \]
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Add 11 to both sides: \[ y = -\frac{8}{5}x + \frac{32}{5} + 11 \]
To combine the constant terms: \[ 11 = \frac{55}{5} \]
So, \[ y = -\frac{8}{5}x + \frac{32}{5} + \frac{55}{5} \] \[ y = -\frac{8}{5}x + \frac{87}{5} \]
Thus, the equation of the line that passes through the point (4, 11) and is perpendicular to the line \( Y = \frac{5}{8}x + 7 \) is:
\[ y = -\frac{8}{5}x + \frac{87}{5} \]
Since you provided multiple-choice options, it's worth checking which of those matches this derived equation. However, the options you've given seem to be jumbled, and it's unclear which form they are in, so please check your options against the derived equation for clarity.