Why do we subtract the y-values and subtract the x-values to find the slope?

You mean that if we have two pairs of points, (x1, y1) and (x2, y2)

we can find the slope of the line that conects those two points as:

a = (y2 - y1)/(x2 - x1)

remember that a linear equation has the shape:

y(x) = a*x + b

Now let's prove it:

for our case we have that

y1 = a*x1 + b

y2 = a*x2 + b

we can isolate be in both equtaions and get:

b = y1 - a*x1

b = y2 - a*x2

this means that:

y1 - a*x1 = y2 - a*x2

then:

a*x2 - a*x1 = y2 - y1

a*(x2 - x1) = y2 - y1

a = (y2 - y1)/(x2 - x1)

Which is the thing we wanted to get.

2

step-by-step explanation: