Find the distance from point (-6,0) to the line 2x - 3y = -3?

Question

I need it to be explained step by step, including formulas used. I am a 9th grade student.

Answer 1

make the eq = 0

2x-3y+3=0

distance= 2(-6)-3(0)+3/√(2^2)+(-3^2)
distance=-12+3/√4+9
distance=-9/√13
distance=-2.49

Answer 2

I'm guessing you want to find the shortest distance, right?

The line segment describing your distance will be perpendicular. This should be apparent if you draw a picture- any other angle would make this line longer.

I would first rewrite your equation in slope-intercept form:
2x - 3y = -3,
-3y = -2x - 3,
3y = 2x + 3,
y = 2/3 x + 1.

Now we want to characterize the line perpendicular to this which also passes through (-6, 0).
So your slope is 2/3. A line perpendicular to this slope would have a slope equal to the negative reciprocal. Your textbook should explain the reasons why this is so.
The slope for our new equation is -3/2, so the equation will look like:
y = -3/2 x + b.
We know this line will pass through (-6, 0), so let x = -6 and y = 0
and solve for 'b':
0 = -3/2 * -6 + b,
0 = 9 + b,
b = -9.

Thus the equation of the line where we will measure the distance is: y = -3/2 x - 9.

Now, if we can discover where these lines intersect, we'll have two points with which to determine distance. Set up and solve this system:
y = 2/3 x + 1 and
y = -3/2 x - 9.

Since y = y,
2/3 x + 1 = -3/2 x - 9,
Multiply through by 6 to remove fractions:
4x + 6 = -9x - 54,
13x = -60,
x = -60/13. (you better double-check me on all this)

so y = 2/3 * -60/13 + 1
y = -120/39 + 1
y = -81/39

Now that you have these values, you can set up a right triangle where the hypoteneus is the distance you seek. You can use the Pythagorean Theorem from there.

Feel free to email me for any clarifications.

Answer 3

putting the equation of the line in the standard form
=2x-3y+3=0
distance from the point (-6,0) to the line 2x-3y+3=0
is given by the formula Ax1+By1+C/rt(A^2+B^2)
substituting
distance=2(-6)-3(0)+3/rt(2^2+3^2)
=-12/rt13
ignoring the negative sign and rationalising d=12rt13/13 units

Answer 4

3y=2x+3
y=2/3x+1

distance =1^2+6^2=(sqaure root answer)=6.082