Question

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

Answer 1

(-3,5)(6,-1)
slope = (-1 - 5) / (6 - (-3) = -6/9 which reduces to - 2/3 <=

Answer 2

The slope of the line containing (-3,5) and (6,-1) is
`-(2)/(3)`

Explanation:

Given a line which is writing in the following way :

`y=ax+b`

We define the slope as the number ''a''.

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.

`(-3,5)=(x1,y1)\\(6,-1)=(x2,y2)`

For the first point we get the equation (I)

`y1=ax1+b\\5=a(-3)+b`

`5=-3a+b` (I)

For the second point we get the equation (II)

`y2=ax2+b\\-1=a6+b`

`-1=6a+b` (II)

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

`-1=6a+b\\b=-1-6a`

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

`5=-3a+b\\5=-3a+(-1-6a)\\5=-3a-1-6a`

`6=-9a\\a=-(6)/(9)=-(2)/(3)`

`a=-(2)/(3)`

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

`b=-1-6a`

`b=-1-6(-(2)/(3))\\ b=-1+4=3`

`b=3`

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

`y=ax+b`

`y=-(2)/(3)x+3`

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

`5=-(2)/(3)(-3)+3 \\5=2+3\\5=5`

And for the other point :

`-1=-(2)/(3)(6)+3\\-1=-4+3\\-1=-1`

We answer that the slope is
`-(2)/(3)` given that
`-(2)/(3)` is multiplying the variable ''x'' in the equation.

We also could use the following formula :

The slope of a line containing two points
`A=(x1,y1)` and

`B=(x2,y2)` 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 =
`(-6)/(9)=-(2)/(3)` which is the answer we obtained using the another method.