Question

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

Answer 1

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`.