# 11.4 Understand slope of a line  (Page 4/9)

 Page 4 / 9
 What is the rise? The rise is 2. What is the run? The run is 0. What is the slope? $m=\frac{\text{rise}}{\text{run}}$ $m=\frac{2}{0}$

But we can’t divide by $0.$ Division by $0$ is undefined. So we say that the slope of the vertical line $x=3$ is undefined. The slope of all vertical lines is undefined, because the run is $0.$

## Slope of a vertical line

The slope of a vertical line    , $x=a,$ is undefined.

Find the slope of each line:

1. $\phantom{\rule{0.2em}{0ex}}x=8\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}y=-5$

## Solution

$\phantom{\rule{0.2em}{0ex}}x=8$

This is a vertical line, so its slope is undefined.

$\phantom{\rule{0.2em}{0ex}}y=-5$

This is a horizontal line, so its slope is $0.$

Find the slope of the line: $x=-4.$

undefined

Find the slope of the line: $y=7.$

0

## Use the slope formula to find the slope of a line between two points

Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.

Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair     $\left(x,y\right)$ gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol $\left(x,y\right)$ be used to represent two different points?

Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.

• $\left({x}_{1},{y}_{1}\right)\phantom{\rule{0.2em}{0ex}}\text{read}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sub}\phantom{\rule{0.2em}{0ex}}1,y\phantom{\rule{0.2em}{0ex}}\text{sub}\phantom{\rule{0.2em}{0ex}}1$
• $\left({x}_{2},{y}_{2}\right)\phantom{\rule{0.2em}{0ex}}\text{read}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sub}\phantom{\rule{0.2em}{0ex}}2,y\phantom{\rule{0.2em}{0ex}}\text{sub}\phantom{\rule{0.2em}{0ex}}2$

We will use $\left({x}_{1},{y}_{1}\right)$ to identify the first point and $\left({x}_{2},{y}_{2}\right)$ to identify the second point. If we had more than two points, we could use $\left({x}_{3},{y}_{3}\right),\left({x}_{4},{y}_{4}\right),$ and so on.

To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points $\left(2,3\right)$ and $\left(7,6\right)$ in [link] .

Since we have two points, we will use subscript notation.

$\stackrel{{x}_{1},{y}_{1}}{\left(2,3\right)}\phantom{\rule{1.0em}{0ex}}\stackrel{{x}_{2},{y}_{2}}{\left(7,6\right)}$

On the graph, we counted the rise of $3.$ The rise can also be found by subtracting the $y\text{-coordinates}$ of the points.

$\begin{array}{c}{y}_{2}-{y}_{1}\\ 6-3\\ 3\end{array}$

We counted a run of $5.$ The run can also be found by subtracting the $x\text{-coordinates}.$

$\begin{array}{c}{x}_{2}-{x}_{1}\\ 7-2\\ 5\end{array}$
 We know $m=\frac{\text{rise}}{\text{run}}$ So $m=\frac{3}{5}$ We rewrite the rise and run by putting in the coordinates. $m=\frac{6-3}{7-2}$ But 6 is the $y$ -coordinate of the second point, ${y}_{2}$ and 3 is the $y$ -coordinate of the first point ${y}_{1}$ . So we can rewrite the rise using subscript notation. $m=\frac{{y}_{2}-{y}_{1}}{7-2}$ Also 7 is the $x$ -coordinate of the second point, ${x}_{2}$ and 2 is the $x$ -coordinate of the first point ${x}_{2}$ . So we rewrite the run using subscript notation. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

We’ve shown that $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ is really another version of $m=\frac{\text{rise}}{\text{run}}.$ We can use this formula to find the slope of a line when we have two points on the line.

## Slope formula

The slope of the line between two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ is

$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

$\text{Slope is}\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}\text{of the second point minus}\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}\text{of the first point}$
$\text{over}$
$x\phantom{\rule{0.2em}{0ex}}\text{of the second point minus}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{of the first point.}$

Doing the Manipulative Mathematics activity “Slope of Lines Between Two Points” will help you develop a better understanding of how to find the slope of a line between two points.

Find the slope of the line between the points $\left(1,2\right)$ and $\left(4,5\right).$

## Solution

 We’ll call $\left(1,2\right)$ point #1 and $\left(4,5\right)$ point #2. $\stackrel{{x}_{1},{y}_{1}}{\left(1,2\right)}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\stackrel{{x}_{2},{y}_{2}}{\left(4,5\right)}$ Use the slope formula. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Substitute the values in the slope formula: $y\phantom{\rule{0.2em}{0ex}}$ of the second point minus $\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}$ of the first point $m=\frac{5-2}{{x}_{2}-{x}_{1}}$ $x\phantom{\rule{0.2em}{0ex}}$ of the second point minus $\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}$ of the first point $m=\frac{5-2}{4-1}$ Simplify the numerator and the denominator. $m=\frac{3}{3}$ $m=1$

Let’s confirm this by counting out the slope on the graph.

The rise is $3$ and the run is $3,$ so

$\begin{array}{}\\ m=\frac{\text{rise}}{\text{run}}\hfill \\ m=\frac{3}{3}\hfill \\ m=1\hfill \end{array}$

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?