Question 131771
The slope is given by {{{(y[1]-y[2])/(x[1]-x[2])}}}.  Just decide which point will be 1 and which will be 2 (it doesn't matter), and plug in the numbers.  If (2,5) is point 1, then {{{x[1]=2}}}, and {{{y[1]=5}}}.


Given two points you need to use the two point form of the equation of a line:


{{{y-y[1]=((y[1]-y[2])/(x[1]-x[2]))(x-x[1])}}}


Again, just plug in the numbers and do the arithmetic


{{{y-5=((5-8)/(2-4))(x-2)}}}
{{{y-5=((-3)/(-2))(x-2)}}}
{{{y-5=(3/2)(x-2)}}}
{{{2y-10=3(x-2)}}}
{{{2y-10=3x-6}}}
{{{-3x+2y=-6+10}}}
{{{3x-2y=-4}}} is the equation in standard ({{{ax+by=c}}} form.


For your other problem, it is done exactly the same way.  The rate of change is the same as the slope.  Your points are (3,10) and (12,55).  Use {{{(y[1]-y[2])/(x[1]-x[2])}}} to calculate the slope.