What is the slope of the line containing (–3, 5) and (6, –1)

First we find the equation of the line containing (-3,5) and (6,-1). We do this by replacing in the line equation the variables ''x'' and ''y'' by the values of the points.

[tex](-3,5)=(x1,y1)\\(6,-1)=(x2,y2)[/tex]

For the first point we get the equation (I)

[tex]y1=ax1+b\\5=a(-3)+b[/tex]

[tex]5=-3a+b[/tex] (I)

For the second point we get the equation (II)

[tex]y2=ax2+b\\-1=a6+b[/tex]

[tex]-1=6a+b[/tex] (II)

Now we replace one parameter in terms of the other.For example, in (II) we replace b in terms of a :

[tex]-1=6a+b\\b=-1-6a[/tex]

Now we replace ''b'' in the equation (I)

[tex]5=-3a+b\\5=-3a+(-1-6a)\\5=-3a-1-6a[/tex]

[tex]6=-9a\\a=-\frac{6}{9}=-\frac{2}{3}[/tex]

[tex]a=-\frac{2}{3}[/tex]

With this value of a we replace in the equation of ''b'' :

[tex]b=-1-6a[/tex]

[tex]b=-1-6(-\frac{2}{3})\\ b=-1+4=3[/tex]

[tex]b=3[/tex]

Now we write the equation of the line containing both points :

[tex]y=ax+b[/tex]

[tex]y=-\frac{2}{3}x+3[/tex]

We verify that actually this line contains both points by replacing ''x'' and ''y'' :

[tex]5=-\frac{2}{3}(-3)+3 \\5=2+3\\5=5[/tex]

And for the other point :

[tex]-1=-\frac{2}{3}(6)+3\\-1=-4+3\\-1=-1[/tex]

We answer that the slope is [tex]-\frac{2}{3}[/tex] given that [tex]-\frac{2}{3}[/tex] is multiplying the variable ''x'' in the equation.

We also could use the following formula :

The slope of a line containing two points [tex]A=(x1,y1)[/tex] and

[tex]B=(x2,y2)[/tex] is equal to Δy / Δx

Δy = y2 - y1 and Δx = x2 - x1

or also

Δy = y1 - y2 and x1 - x2

For example for the points (-3,5) and (6,-1) :

Δy = -1 - 5 = - 6

and Δx = 6 - (-3) = 9

⇒ Δy / Δx = [tex]\frac{-6}{9}=-\frac{2}{3}[/tex] which is the answer we obtained using the another method.