To find the equation of the line that is perpendicular to the line given by \( y = -\frac{3}{4}x + 7 \) and passes through the point \( (6, 1) \), we first need to determine the slope of the perpendicular line.
The slope of the given line is \( -\frac{3}{4} \). The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope \( m \) of the perpendicular line will be:
\[ m = -\frac{1}{-\frac{3}{4}} = \frac{4}{3} \]
Now that we have the slope of the perpendicular line, we can use the point-slope form of the line equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
Where \( (x_1, y_1) \) is the point through which the line passes. Here, \( (x_1, y_1) = (6, 1) \) and \( m = \frac{4}{3} \). Substituting these values into the equation gives:
\[ y - 1 = \frac{4}{3}(x - 6) \]
Next, we can simplify this equation:
- Distribute \( \frac{4}{3} \):
\[ y - 1 = \frac{4}{3}x - \frac{4}{3} \cdot 6 \] \[ y - 1 = \frac{4}{3}x - 8 \]
- Add 1 to both sides to solve for \( y \):
\[ y = \frac{4}{3}x - 8 + 1 \] \[ y = \frac{4}{3}x - 7 \]
Thus, the equation of the line that is perpendicular to \( y = -\frac{3}{4}x + 7 \) and goes through the point \( (6, 1) \) is:
\[ y = \frac{4}{3}x - 7 \]