Question

Write a slope-intercept equation for a line passing through the point (6, 12) that is parallel to y = 1/2x + 1. Then write a second equation for a line passing through the given point that is perpendicular to the given line. Which answer below is correct?

1) Parallel: y = 1/2x + 12, Perpendicular: y = -2x + 12
2) Parallel: y = 1/2x + 9, Perpendicular: y = 2x + 24
3) Parallel: y = 1/2x + 9, Perpendicular: y = -2x + 24

Answer 1

To find the equation of a line parallel to y = 1/2x + 1 passing through the point (6, 12), use the same slope and solve for the y-intercept. The equation is y = 1/2x + 9. For a line perpendicular to y = 1/2x + 1, find the negative reciprocal of the slope and use the point-slope form of the equation. The equation is y = -2x + 24.

Step-by-step explanation:

To find the equation of a line parallel to y = 1/2x + 1, we need to use the same slope because parallel lines have the same slope. Since the slope of the given line is 1/2, the equation will be y = 1/2x + b. To find the value of b, we can substitute the coordinates of the given point (6, 12) into the equation and solve for b. Substituting x = 6 and y = 12 into the equation, we get 12 = (1/2)(6) + b. Solving for b, we find that b = 9. Therefore, the equation of the line parallel to y = 1/2x + 1 passing through the point (6, 12) is y = 1/2x + 9.

For a line perpendicular to y = 1/2x + 1, we need to find a slope that is negative reciprocal to the slope of the given line. The slope of y = 1/2x + 1 is 1/2, so the slope of the perpendicular line will be -2. Using the point-slope form, the equation will be y - 12 = -2(x - 6). Simplifying this equation, we get y = -2x + 24. Therefore, the correct answer is:

Parallel: y = 1/2x + 9

Perpendicular: y = -2x + 24