Question 696105
If you have the equation for a line in the form {{{y=mx+b}}},
called the {{{highlight(slope-intercept)}}} form
(or if you can transform the equation you have into that form),
the numbers {{{m}}} and {{{b}}} are the slope and y-intercept respectively.
For {{{y=5x+3}}} ,  {{{highlight(slope=5)}}} and {{{highlight(y-intercept=3)}}}
For {{{y=-8x-1}}} ,  {{{highlight(slope=-8)}}} and {{{highlight(y-intercept=-1)}}}
For {{{y=4+2x}}}<-->{{{y=2x+4}}} ,  {{{highlight(slope=2)}}} and {{{highlight(y-intercept=4)}}}
 
If you have two points, you can determine the slope and, the y-intercept, and the equation of the line.
I see two simple ways about it:
solve it as a system of equations,
or use what you learned about analytical geometry.
The choice of method depends on your preference (what is easier for you) and your teacher's preference.
 
USING SYSTEMS OF EQUATIONS:
For the line that passes through (8,4) and (2,1),
substituting {{{x=8}}} and {{{y=4}}} (coordinates of (8,4)) into {{{y=mx+b}}} we get
{{{4=8m+b}}}
Doing the same with point (2,1), we get
{{{1=2m+b}}}
That gives you the system {{{system(4=8m+b,2x+b)}}}
that you can solve for {{{m}}} and {{{b}}}.
Subtracting the second equation from the first, you get
{{{3=6m}}} --> {{{m=3/6=1/2}}} so {{{highlight(slope=1/2)}}}
Substituting that value into {{{1=2m+b}}} , you get
{{{1=2(1/2)+b}}} --> {{{1=1+b}}} --> {{{b=0}}} so {{{highlight(y-intercept=0)}}}
 
USING ANALYTICAL GEOMETRY:
The slope of a line is defined as the ratio between the increase in y-coordinate and the increase in x-coordinate when going from one point to the other.
That is easier to write in words than as the usual cumbersome formula, which looks like
{{{slope=highlight(m=(y[1]-y[2])/(x[1]-x[2]))}}}
for known points {{{P[1]}}} ({{{x[1]}}},{{{y[1]}}}) and {{{P[1]}}} ({{{x[1]}}},{{{y[1]}}}).
Then, for any point (x,y) on the line, and a known point ({{{x[A]}}},{{{y[A]}}})
{{{m=(y-y[A])/(x-x[A])}}} --> {{{highlight(y-y[A]=m(x-x[A]))}}}
gives the equation of the line in the {{{highlight(point-slope)}}} form,
which can be transformed into the {{{highlight(slope-intercept)}}} form.
 
For the line that passes through (8,4) and (2,1),
so as not to get all those numbers mixed up,
you could write into the formula the coordinates of one point at a time.
You may start with point (8,4), with {{{x=8}}} and {{{y=4}}}, and write
{{{slope=(4-y[2])/(8-x[2])}}} (leaving the spaces for {{{x[2]}}} and {{{y[2]}}} blank).
Then, you could fill in the coordinates of the other point to get
{{{slope=(4-1)/(8-2)=3/6=1/2}}} so {{{highlight(slope=1/2)}}}
Then you can write the equation of the line in point-slope form,
maybe using point (2,1) as your point:
{{{y-1=(1/2)(x-2)}}}
That can be transformed int the slope intercept form:
{{{y-1=(1/2)(x-2)}}}-->{{{y-1=(1/2)x-1)}}}-->{{{highlight((1/2)x)}}}
The y-intercept is that invisible {{{0}}} added after the {{{(1/2)x}}}
so {{{highlight(y-intercept=0)}}}.
 
For the line that passes through (-9,8) and (10,- 4)
Slope={{{(8-(-4))/(-9-10)=12/(-19)=highlight(-12/19)}}}
{{{y-(-4)=(-12/19)(x-10)}}}-->{{{y+4=(-12/19)x+120/19}}}-->{{{y=(-12/19)x+120/19-4}}}-->{{{y=(-12/19)x+120/19-76/19}}}-->{{{y=(-12/19)x+44/19}}}
So {{{highlight(y-intercept=44/19)}}}.
 
For the line that passes through (-1,-2) and (-3,-4)
Slope={{{(-2-(-4))/(-1-(-3))=(-2+4)/(-1+3)=2/2=highlight(1)}}}
{{{y-(-4)=1(x-(-3))}}}-->{{{y+4=x+3}}}-->{{{y=x+3-4}}}-->{{{y=x-1}}}
So {{{highlight(y-intercept=-1)}}}.