Question

Use the given conditions to write an equation for the line in​ point-slope form and in​ general form.

Answer 1

First, simplify the equation given into slope intercept form.

slope intercept form is y = mx + b, where b is the y intercept and m is the slope.

3x - 2y - 5 = 0

I would move the 2y to the other side.

3x - 5 = 2y

Then, since y can't have a coefficient, divide everything by 2;

y = 3/2x - 5/2

So there's your slope intercept form.

Point slope form is:

$y - y_{1} = m( x} - x_{1} )$

Where those with a subscript of 1 are part of the same point.

So you already known one point; -8, 8. I'll just do that as the pair with subscript 1. You know the slope as well from the slope intercept form; 3/2.

You can just plug those in. 

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

Now to change this to general form.

First, distribute 3/2 to x and 8.

y - 8 = 3/2x + 12
-3/2x + y = 20
-3/2x + y - 20 = 0

Since there are no fractions, multiply everything by 2.

-3x + 2y - 40 = 0

Answer 2

First, let's convert the given line equation into slope-intercept form to find the slope.
Let's start converting by subtracting both sides by 3x.

$-2y-5=-3x$

Add both sides by 5.

$-2y=-3x+5$

Divide both sides by -2.

$y= \frac{3}{2} x- \frac{5}{2}$

The slope of the equation given is 3/2. Since the point-slope form line is parallel to that, the point-slope form equation must also have a slope of 3/2.

Now, let's use point-slope form.

For a line with slope m and that passes through $(x_1,y_1)$, the point slope form equation is the following:

$y-y_1=m(x-x_1)$

We know the passing point and the slope. Now, let's plug them into the point-slope form formula.

$y-8= \frac{3}{2} (x-(-8))$
$y-8= \frac{3}{2} (x+8)$

That is your answer for the point-slope form equation.
To change this to general form, distribute first.

$y-8= \frac{3}{2} x+12$

Subtract both sides by $\frac{3}{2} x$ and 12.

$- \frac{3}{2} x+y-20=0$

Multiply both sides by -2

$3x-2y+40=0$

That's your answer for the general form equation.
I hope this helps and have an awesome day! :)