Final answer:
The equation of the line that passes through the point (-2,14) and is perpendicular to the line y = 1/2x - 3 is y = -2x + 10.
Step-by-step explanation:
To find the equation of a line that is perpendicular to a given line, we need to determine the slope of the given line and then find the negative reciprocal of that slope. Let's find the slope of each given line:
a. y = 2x + 7: The slope is 2.
b. y = -2x - 7: The slope is -2.
c. y = -\frac{1}{2}x + 3: The slope is -\frac{1}{2}.
d. y = \frac{1}{2}x - 3: The slope is \frac{1}{2}.
Since the line is perpendicular to the given line, the slope of the perpendicular line will be the negative reciprocal of the slope of the given line. The given line has a slope of \frac{1}{2}, so the perpendicular line will have a slope of -2.
Now, we can use the point-slope form of a linear equation (y - y1 = m(x - x1)) to find the equation of the perpendicular line. Using the point (-2, 14), we have:
y - 14 = -2(x - (-2))
y - 14 = -2(x + 2)
y - 14 = -2x - 4
y = -2x + 10
Therefore, the equation of the line that passes through the point (-2, 14) and is perpendicular to the line y = \frac{1}{2}x - 3 is y = -2x + 10.