To determine which line is perpendicular to the line \( y = -\frac{7}{4}x \) and goes through the point \((-7, 5)\), we first need to find the slope of the given line and then identify the slope of the line that would be perpendicular to it.
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Identify the slope of the given line: The slope of the line \( y = -\frac{7}{4}x \) is \( -\frac{7}{4} \).
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Determine the slope of the perpendicular line: The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \(-\frac{7}{4}\) is: \[ \frac{4}{7} \]
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Use the point-slope form to find the equation of the perpendicular line: Using the point \((-7, 5)\) and the slope \(\frac{4}{7}\), we can use the point-slope form: \[ y - y_1 = m(x - x_1) \] Substitute \( (x_1, y_1) = (-7, 5) \) and \( m = \frac{4}{7} \): \[ y - 5 = \frac{4}{7}(x + 7) \]
To write it in slope-intercept form (y = mx + b), we can expand and simplify: \[ y - 5 = \frac{4}{7}x + 4 \] \[ y = \frac{4}{7}x + 9 \]
Now we look at the provided responses:
- \( y = \frac{4}{7}x + 9 \)
- \( y = -\frac{4}{7}x + 1 \)
- \( y = -\frac{7}{4}x + 5 \)
- \( y = \frac{4}{7}x + 5 \)
The first option, \( y = \frac{4}{7}x + 9 \), is the correct line that is perpendicular to \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\).
Thus, the answer is: y = \frac{4}{7}x + 9.