# Linear equations  (Page 2/6)

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The graph of the line $y=b$ , where $b$ is a constant, is a horizontal line that passes through the point (0, $b$ ). Every point on this line has the y-coordinate $b$ , regardless of the x-coordinate.

Graph the lines: $x=-2$ , and $y=3$ .

The graph of the line $x=-2$ is a vertical line that has the x-coordinate –2 no matter what the y-coordinate is. Therefore, the graph is a vertical line passing through (–2, 0).

The graph of the line $y=3$ , is a horizontal line that has the y-coordinate 3 regardless of what the x-coordinate is. Therefore, the graph is a horizontal line that passes through (0, 3).

Most students feel that the coordinates of points must always be integers. This is not true, and in real life situations, not always possible. Do not be intimidated if your points include numbers that are fractions or decimals.

## Section overview

In this section, you will learn to:

1. Find the slope of a line if two points are given.
2. Graph the line if a point and the slope are given.
3. Find the slope of the line that is written in the form $y=\text{mx}+b$ .
4. Find the slope of the line that is written in the form $\text{Ax}+\text{By}=c$ .

In the last section, we learned to graph a line by choosing two points on the line. A graph of a line can also be determined if one point and the "steepness" of the line is known. The precise number that refers to the steepness or inclination of a line is called the slope of the line.

From previous math courses, many of you remember slope as the "rise over run," or "the vertical change over the horizontal change" and have often seen it expressed as:

$\frac{\text{rise}}{\text{run}},\frac{\text{vertical change}}{\text{horizontal change}},\frac{\mathrm{\Delta y}}{\mathrm{\Delta x}}\text{etc.}$

We give a precise definition.

If ( ${x}_{1}$ , ${y}_{1}$ ) and ( ${x}_{2}$ , ${y}_{2}$ ) are two different points on a line, then the slope of the line is

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

Find the slope of the line that passes through the points (-2, 3) and (4, -1), and graph the line.

Let $\left({x}_{1},{y}_{1}\right)=\left(-2,3\right)$ and $\left({x}_{2},{y}_{2}\right)=\left(4,-1\right)$ then the slope

$m=\frac{-1-3}{4-\left(-2\right)}=-\frac{4}{6}=-\frac{2}{3}$

To give the reader a better understanding, both the vertical change, –4, and the horizontal change, 6, are shown in the above figure.

When two points are given, it does not matter which point is denoted as $\left({x}_{1},{y}_{1}\right)$ and which $\left({x}_{2},{y}_{2}\right)$ . The value for the slope will be the same. For example, if we choose $\left({x}_{1},{y}_{2}\right)=\left(4,-1\right)$ and $\left({x}_{2},{y}_{2}\right)=\left(-2,3\right)$ , we will get the same value for the slope as we obtained earlier. The steps involved are as follows.

$m=\frac{3-\left(-1\right)}{-2-4}=\frac{4}{-6}=-\frac{2}{3}$

The student should further observe that if a line rises when going from left to right, then it has a positive slope; and if it falls going from left to right, it has a negative slope.

Find the slope of the line that passes through the points (2, 3) and (2, -1), and graph.

Let $\left({x}_{1},{y}_{1}\right)=\left(2,3\right)$ and $\left({x}_{2},{y}_{2}\right)=\left(2,-1\right)$ then the slope

$m=\frac{-1-3}{2-2}=-\frac{4}{0}=\text{undefined}$

The slope of a vertical line is undefined.

Graph the line that passes through the point (1, 2) and has slope $-\frac{3}{4}$ .

Slope equals $\frac{\text{rise}}{\text{run}}$ . The fact that the slope is $\frac{-3}{4}$ , means that for every rise of –3 units (fall of 3 units) there is a run of 4. So if from the given point (1, 2) we go down 3 units and go right 4 units, we reach the point (5, –1). The following graph is obtained by connecting these two points.

Alternatively, since $\frac{3}{-4}$ represents the same number, the line can be drawn by starting at the point (1,2) and choosing a rise of 3 units followed by a run of –4 units. So from the point (1, 2), we go up 3 units, and to the left 4, thus reaching the point (–3, 5) which is also on the same line. See figure below.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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