Write an equation for the line perpendicular to line XY that contains point Z.

line XY: 3x + 2y = -6, Z(3,2)

I realize that the slope of a perpendicular line is the opposite reciprocal of the first line. But, how do I use point Z in this problem? Thank you.

y= -3/2x-3

so the new slope will be 2/3
so then y=2/3 x+b
put point z into the equation, and solve for b.

Since you know that the slopes of perpendicular lines are opposite reciprocals, you should accept that the new equation must be

2x - 3y = c
plug in the point
6 - 4 = c = 2

so 2x - 3y = 2

Well, well, well, trying to make math more interesting, huh? I've got just the equation for you, my friend. Hang on tight!

First, let's determine the slope of line XY. We can rewrite the equation in slope-intercept form (y = mx + b), where m represents the slope. So, rearranging the equation, we have: y = (-3/2)x - 3.

Now, the perpendicular line's slope will be the opposite reciprocal of the slope of line XY. Flipping the fraction and changing the sign, we get 2/3. Are you still with me?

To find the equation, we can plug the new slope (2/3) and the coordinates of point Z (3,2) into the point-slope form (y - y1 = m(x - x1)). So, substituting the values, we have:

y - 2 = (2/3)(x - 3).

And there you have it! The equation of the line perpendicular to line XY that contains point Z is y - 2 = (2/3)(x - 3). I hope that puts a smile on your face! Let me know if you need any more chuckles, I mean, assistance.

To find the equation of the line perpendicular to line XY and passing through point Z, you can follow these steps:

Step 1: Determine the slope of line XY.
The equation of line XY is given as 3x + 2y = -6. To find the slope, rearrange the equation into slope-intercept form (y = mx + b), where m is the slope:
2y = -3x - 6
y = (-3/2)x - 3
Comparing with the general form y = mx + b, you can determine that the slope of line XY is -3/2.

Step 2: Find the slope of the line perpendicular to XY.
The slope of a line perpendicular to another line is the negative reciprocal of its slope. In this case, the slope of the line perpendicular to XY is 2/3 (negative reciprocal of -3/2).

Step 3: Use the slope and the point Z to write the equation.
Since you have the slope (2/3) and the point Z(3,2), you can use the point-slope form of a line to write the equation. The point-slope form is given as: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Plugging in the values, you get:
y - 2 = (2/3)(x - 3)
Simplifying, you obtain:
y - 2 = (2/3)x - 2
y = (2/3)x

Therefore, the equation of the line perpendicular to XY and passing through point Z(3,2) is y = (2/3)x.