Question

How do you write in point slope form an equation of the line through each pair of points (2,7) and (-4,1)?

Answer 1

Two points uniquely identify a line, and the formula we must use is the following: when given two points `(x_1,y_1)` and `(x_2,y_2)`, the line passing through both of them is

`frac{y-y_2}{y_1-y_2}=\frac{x-x_2}{x_1-x_2}`

Plugging the values `(x_1,y_1)=(2,7)` and `(x_2,y_2)=(-4,1)` into that formula, we get

`\frac{y-1}{7-1}=\frac{x+4}{2+4} \iff \frac{y-1}{6} = \frac{x+4}{6} \iff y-1 = (x+4)`

Which can be brought into the standard form:

`y-1 = x+4 \iff y = x+5`

You can easily check that this is the right solution, because for both points `(2,7)` and `(-4,1)` you have that the `y` coordinate equals the `x` one plus `5`.