Final answer:
To find the y-intercept of a perpendicular line through the point (2, 7) to y = -3/2x + 6, we determine the slope of the perpendicular line as 2/3, use the point-slope form to find its equation, and calculate the y-intercept as 10/3 or approximately 3.33.
Step-by-step explanation:
The question is asking to find the y-intercept of a line that is perpendicular to the given line y = -3/2x + 6 and passes through the point (2, 7). First, we determine the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line. Since the slope of the given line is -3/2, the slope of the perpendicular line will be 2/3. Using the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point through which the line passes, we can find the equation of the perpendicular line:
y - 7 = (2/3)(x - 2)
Expanding and simplifying this, we have:
y = (2/3)x + 14/3 - 4/3
y = (2/3)x + 10/3
So, the y-intercept of our perpendicular line is 10/3 or approximately 3.33. This is the point where the line crosses the y-axis, which can be written as the point (0, 10/3).