Question 135826
Let's first identify your points:

{{{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}}}


Add 5 to both sides:
{{{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


Both true, answer checks.


Do the rest of your problems the same way.  Hint: Be extremely cautious with signed numbers.  Another hint:  Check your work for every problem.  If an answer doesn't check, the odds are good that you made a sign error.  Good luck.


Super-Double-Plus-Extra Credit.  What would have happened if I had identified the points the opposite way in the beginning of this problem, that is, if I had said (1,5) is {{{P[2]}}} and (2,7) is {{{P[1]}}}?  Would I have gotten the same or different results at the end?