Question

What are the three standard forms of equation of a straight line? Also find the

Answer 1

There are a few more than three, so they're really asking for the three they taught you already.

Usually we start with slope-intercept form, slope m, y intercept b

y = mx + b

The form in the question is called standard form or Cartesian form.   Ironically, there are two different standard forms:

ax + by + c  = 0

ax + by = c

They're more or less the same; pick the one your teacher likes.  Next comes point-slope form.  The line through point (a,b) with slope m is

y - b = m(x-a)

There's also point-point form.  Given two points on the line (a,b) and (c,d) the line that joins them is

(c-a)(y-b)=(d-b)(x-a)

That's about it for the non-parametric forms.  There are also parametric forms which generalize to higher dimensions.

The parametric point-point form says the line joining (a,b) and (c,d) is

(x,y)=(1-t)(a,b) + t(c,d)

t is a parameter which sweeps over the reals to generate the line.  When the linear factors add up to one like that, it's called an affine combination.  

Similarly, given a point (a,b) and a direction vector (m,n) we can write the line in point-direction vector form.

(x,y)=(a,b) + t(m,n)

I could probably come up with one or two more, but those are the main ones.

What's the slope of

$ax + by + c=0$

We solve for y:

$by =-ax - c$

$y = - \frac a b x - \frac c b$

That line has slope -a/b.  Of course if b is zero we say there's no slope.

Standard form handles the vertical line case here, so is often superior to the slope forms.