What is the equation of the line passing through (4, 2) and perpendicular to the line passing through the points (9,7) and (11,4)?

Here's what I did:

(7-4)/(9-11) = -3/2

y-2= (-2/3)(x-4)
y-2=(-2x/3) + (8/3)
y = (-2x/3) + 14/3

but the answer key says that it's y=(2x/3) - (2/3)
did I make a mistake?

since the line is perpendicular, their slopes are negative reciprocals of each other, that means, they must be OPPOSITE in sign

so first slope = -3/2
2nd slope = +2/3

y-2 = (2/3)(x-4)
3y-6 = 2x-8
3y = 2x - 2
y = (2/3)x - 2/3

oh yeah thanks I forgot that the reciprocal had to be negative

To find the equation of the line passing through (4, 2) and perpendicular to the line passing through (9, 7) and (11, 4), you need to follow these steps:

1. Find the slope of the line passing through (9, 7) and (11, 4).
- The slope of a line passing through two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).

Using the coordinates (9, 7) and (11, 4):
slope = (4 - 7) / (11 - 9)
= -3 / 2

2. For a line to be perpendicular to another line, their slopes must be negative reciprocals of each other.
- The negative reciprocal of a slope 'm' is -1/m.

Since the slope of the given line is -3/2, the slope of the line perpendicular to it is:
perpendicular_slope = -1 / (-3/2)
= 2/3

3. Use the point-slope form of a line to find the equation of the line passing through (4, 2) with the perpendicular slope of 2/3.
- The point-slope form is given by the equation: y - y1 = m (x - x1), where (x1, y1) is any point on the line and m is the slope.

Plugging in the values (4, 2) for (x1, y1) and 2/3 for m:
y - 2 = (2/3)(x - 4)

4. Simplify the equation to get it in the slope-intercept form (y = mx + b).
Distribute (2/3) to (x - 4):
y - 2 = (2/3)x - (8/3)

Move -2 to the other side of the equation:
y = (2/3)x - (8/3) + 2

Combine like terms and convert to a common denominator:
y = (2/3)x - (8/3) + 6/3

Simplify further:
y = (2/3)x - 2/3

Therefore, the equation of the line passing through (4, 2) and perpendicular to the line passing through (9, 7) and (11, 4) is y = (2/3)x - 2/3, matching the answer key.

It seems like you made a mistake when distributing (-2/3) to (x - 4). Double-check your calculations from that point onwards to find the discrepancy.