Which equation represents the line that passes through the point \( (2,-7) \) and is perpendicular to the line whose equation is \( y=\frac{3}{4} x+4 \) ?

To find the equation of the line that passes through the point \( (2,-7) \) and is perpendicular to the line \( y = \frac{3}{4} x + 4 \), begin by determining the slope of the given line, which is \( \frac{3}{4} \). The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is \( -\frac{4}{3} \). Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the given point \( (2, -7) \) and \( m \) is the new slope \( -\frac{4}{3} \), we can substitute these values in: \[ y - (-7) = -\frac{4}{3}(x - 2) \] Simplifying this equation will yield the desired line's equation. Calculating this step yields: \[ y + 7 = -\frac{4}{3}x + \frac{8}{3} \] Now, converting this into slope-intercept form \( y = mx + b \): \[ y = -\frac{4}{3}x + \frac{8}{3} - 7 \] \[ y = -\frac{4}{3}x - \frac{13}{3} \] This gives us the equation of the line that is perpendicular to the given line and passes through the point \( (2, -7) \): \[ y = -\frac{4}{3}x - \frac{13}{3} \]