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

Answer:

The slope of the line containing (-3,5) and (6,-1) is $-\frac{2}{3}$

Step-by-step 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)

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

$6=-9a\\a=-\frac{6}{9}=-\frac{2}{3}$

$a=-\frac{2}{3}$

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

$b=-1-6a$

$b=-1-6(-\frac{2}{3})\\ b=-1+4=3$

$b=3$

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

$y=ax+b$

$y=-\frac{2}{3}x+3$

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

$5=-\frac{2}{3}(-3)+3 \\5=2+3\\5=5$

And for the other point :

$-1=-\frac{2}{3}(6)+3\\-1=-4+3\\-1=-1$

We answer that the slope is $-\frac{2}{3}$ given that $-\frac{2}{3}$ 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 = $\frac{-6}{9}=-\frac{2}{3}$ which is the answer we obtained using the another method.