Question 553682
EXAMPLE
Let's say you're looking for the equation of a line that passes trough the point (2,1) with slope 1/4. Consider the point given, and a generic point (x,y) on the same line.
{{{drawing(300,300,-1,5,-1,5,
grid(1),
line(-2,0,6,2),
red(circle(2,1,0.1)),
blue(circle(4.4,1.6,0.1)),
locate(2,1,"(2,1)"), locate(4x1.5, "(x,y)")
)}}} According to the definition of slope, {{{(y-1)/(x-2)=1/4}}}
Multiplying both sides by {{{x-2}}} we get
{{{y-1=(1/4)(x-2)}}} <--This is the point-slope equation found.
Applying the distributive property to simplify the product, and solving for {{{y}}}
{{{y-1=(1/4)x-2(1/4)}}} ---> {{{y-1=(1/4)x-(1/2)}}} ---> {{{y=(1/4)x+1-(1/2)}}}
And that simplifies into the slope intercept-form of the equation
{{{y=(1/4)x+1/2}}}
IN GENERAL
The point-slope form of the equation can be derived from the calculation of the slope between any point (x,y) on the line, and the point given.
Solving that point-slope equation for {{{y}}} you find the slope-intercept form:
{{{y=slope*x+intercept}}}