```Question 135826

{{{P[1]}}}:({{{x[1]}}},{{{y[1]}}}):(1,5)
{{{P[2]}}}:({{{x[2]}}},{{{y[2]}}}):(2,7)

That means that {{{x[1]=1}}}, {{{y[1]=5}}}, {{{x[2]=2}}}, and {{{y[2]=7}}}

Step 1: Find the slope:

The slope is given by {{{m=((y[1]-y[2])/(x[1]-x[2]))}}}

Just substitute the coordinate values:
{{{m=((5-7)/(1-2))=(-2)/(-1)=2}}}

Step 2:

Two point form:  {{{y-y[1]=((y[1]-y[2])/(x[1]-x[2]))(x-x[1])}}}

But notice that the big fraction on the right side of the two-point form is the same as the slope calculation, which allows us to write the point-slope form:

{{{y-y[1]=m(x-x[1])}}}

Substituting, remembering that we said {{{m=2}}} in step 1:

{{{y-5=2(x-1)}}}

Note that once you simplify the big fraction in the two-point form, the two-point and point-slope forms give you identical equations.

Step 3: Solve for y.

{{{y-5=2(x-1)}}}

Distribute:
{{{y-5=2x-2}}}

{{{y=2x+3}}}

Check the answer:  If we did this correctly, the x- and y-coordinates of the given points, when substituted into the equation for x and y, will make the equation a true statement.

{{{5=2(1)+3=2+3=5}}} True
{{{7=2(2)+3=4+3=7}}} True