How do you solve #y-y_1=m(x-x_1)# for m?

This equation is the point slope form for a straight line.

#y-y_1=m(x-x_1)#

#m=(y-y_1)/(x-x_1)#

The point slope equation is used to determine the equation for a straight line, given the slope #(m)# and one point on the line, #(x_1, y_1)#. Suppose you have been given a slope of #m=5#, and a point of #x_1=6# and a point of #y_1=2#.

#y-y_1 = m(x-x_1)#

Plug in given values.
#y-2=5(x-6)#

Distribute the 5.
#y-2=5x-30#

Add 2 to both sides of the equation.
#y=5x-28#

In order to find the slope of a line, using two points on the line, you use the equation #m=(y_2-y_1)/(x_2-x_1)#. Notice it is not identical to the first equation you gave, which is because we need two points to determine the slope. Suppose the line goes through points #(x_1,y_1)=(8,10)# and #(x_2,y_2)=(4,2)#.

#m=(y_2-y_1)/(x_2-x_1)=(10-2)/(8-4)=8/4=2#

To find the equation of this line using the equation #y=mx+b#, you need to find the y-intercept, #b#.

#b=y-mx#

Plug in the slope and the x and y values for one of the two points. I will use the point #(8,10)#.

#b=10-(2*8)=10-16=-6#

I get the same answer if I use the other point, #(4,2)#.

#b=2-(2*4)=2-8=-6#

So the equation of this line is #y=2x-6#.