Final answer:
The equation of the line that passes through (2, -6) and is perpendicular to y = \(\frac{2}{3}\)x + 4 is y = -\frac{3}{2}x - 3.
Step-by-step explanation:
The equation of the line that passes through the point (2, -6) and is perpendicular to the line y = \(\frac{2}{3}\)x + 4 can be determined by first finding the slope of the given line and then finding the negative reciprocal of that slope to get the slope of the perpendicular line. Since the slope of the given line is \(\frac{2}{3}\), the slope of the line perpendicular to it will be \(-\frac{3}{2}\). We can then use the point-slope form of the equation of a line, y - y1 = m(x - x1), where (x1, y1) is the point (2, -6) and m is the slope \(-\frac{3}{2}\).
Plugging in the values, we get the equation y + 6 = -\frac{3}{2}(x - 2). Simplifying this, we multiply both sides by 2 to avoid fractions: 2y + 12 = -3(x - 2), which simplifies further to 2y + 12 = -3x + 6. Subtracting 12 from both sides gives us 2y = -3x - 6. Finally, dividing through by 2 to solve for y, we arrive at the equation of the perpendicular line: y = -\frac{3}{2}x - 3.