Determine an equation for the straight line passing through the point (-3,4) and perpendicular to the line 2x + 3y - 6 = 0.
I converted it to y=2 - (2/3)x which works fine.
This means that 0.5 is probably the right angle, to get the point at -3,4 however is the tougher part. I know it is 0.5x + (2/3)x + 7.5 .. But I got that using a graph calculator with a couple trial and errors. I don't know how to get 'b' here.
Yes the slope of the original line is m = -2/3
Therefore the slope of the line you want is m' = -1/m =3/2
sour line is of form
y = (3/2) x + b
put in your point
4 = (3/2)(-3) + b
b= 8/2 + 9/2 = 17/2
so
y = (3/2) x + 17/2
or
2 y - 3x = 17
Very simple method:
Since the new line must be perpendicular to 2x + 3y - 6 = 0,
it must take the form 3x - 2y = C
plug in (-3,4) ----> 3(-3) - 2(4) = C
C = -17
3x - 2y = -17 , all done
Oh that is real easy! Awesome!
To find the equation for the straight line passing through the point (-3,4) and perpendicular to the line 2x + 3y - 6 = 0, you can follow these steps:
Step 1: Determine the slope of the given line.
The given line is 2x + 3y - 6 = 0. Rewrite it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Solve for y:
3y = -2x + 6
y = (-2/3)x + 2
Comparing this with y = mx + b, we see that the slope (m) of the given line is -2/3.
Step 2: Determine the slope of the line perpendicular to the given line.
Two lines are perpendicular if their slopes are negative reciprocals of each other. The slope (m') of the line perpendicular to the given line is the negative reciprocal of -2/3, which can be found by flipping the fraction and changing its sign:
m' = 3/2
Step 3: Use the point-slope form to find the equation of the perpendicular line.
The point-slope form of a straight line is given by y - y₁ = m'(x - x₁), where (x₁, y₁) is a point on the line and m' is the slope of the line. We can substitute the given point (-3,4) and the perpendicular slope (m') into this formula:
y - 4 = (3/2)(x - (-3))
Step 4: Simplify the equation.
Distribute (3/2) to (x - (-3)):
y - 4 = (3/2)x - 9/2
To eliminate fractions, multiply the entire equation by 2:
2y - 8 = 3x - 9
Move the terms around to put the equation in the general form (Ax + By + C = 0):
3x - 2y + 1 = 0
So, the equation of the straight line passing through the point (-3,4) and perpendicular to the line 2x + 3y - 6 = 0 is 3x - 2y + 1 = 0.