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a x + b y = c

is said to be in general form .

We must stipulate that a and b cannot both equal zero at the same time, for if they were we would have

0 x + 0 y = c

0 = c

This statement is true only if c = 0 . If c were to be any other number, we would get a false statement.

Now, we have the following:

The graphing of all ordered pairs that solve a linear equation in two variables produces a straight line.

This implies,

The graph of a linear equation in two variables is a straight line.

From these statements we can conclude,

If an ordered pair is a solution to a linear equations in two variables, then it lies on the graph of the equation.

Also,

Any point (ordered pairs) that lies on the graph of a linear equation in two variables is a solution to that equation.

The intercept method of graphing

When we want to graph a linear equation, it is certainly impractical to graph infinitely many points. Since a straight line is determined by only two points, we need only find two solutions to the equation (although a third point is helpful as a check).

Intercepts

When a linear equation in two variables is given in general from, a x + b y = c , often the two most convenient points (solutions) to fine are called the Intercepts: these are the points at which the line intercepts the coordinate axes. Of course, a horizontal or vertical line intercepts only one axis, so this method does not apply. Horizontal and vertical lines are easily recognized as they contain only one variable. (See Sample Set C .)

A graph of a line sloped down and to the right. The line intersects the x axis at a positive value of x, and the y axis at a positive value of y. The points where the line intersects the axes are labeled x-intercept and y-intercept respectively.

y -Intercept

The point at which the line crosses the y -axis is called the y -intercept . The x -value at this point is zero (since the point is neither to the left nor right of the origin).

x -Intercept

The point at which the line crosses the x -axis is called the x -intercept  and the y -value at that point is zero. The y -intercept can be found by substituting the value 0 for x into the equation and solving for y . The x -intercept can be found by substituting the value 0 for y into the equation and solving for x .

Intercept method

Since we are graphing an equation by finding the intercepts, we call this method the intercept method

Sample set a

Graph the following equations using the intercept method.

y 2 x = 3

To find the y -intercept , let x = 0 and y = b .

b 2 ( 0 ) = 3 b 0 = 3 b = 3

Thus, we have the point ( 0 , 3 ) . So, if x = 0 , y = b = 3 .

To find the x -intercept , let y = 0 and x = a .

0 2 a = 3 2 a = 3 Divide by -2 . a = 3 2 a = 3 2

Thus, we have the point ( 3 2 , 0 ) . So, if x = a = 3 2 , y = 0 .

Construct a coordinate system, plot these two points, and draw a line through them. Keep in mind that every point on this line is a solution to the equation y 2 x = 3 .

A graph of a line passing through two points with coordinates zero, negative three and three over two, zero.

Got questions? Get instant answers now!

2 x + 3 y = 3

To find the y -intercept , let x = 0 and y = b .

2 ( 0 ) + 3 b = 3 0 + 3 b = 3 3 b = 3 b = 1

Thus, we have the point ( 0 , 1 ) . So, if x = 0 , y = b = 1 .

To find the x -intercept , let y = 0 and x = a .

2 a + 3 ( 0 ) = 3 2 a + 0 = 3 2 a = 3 a = 3 2 a = 3 2

Thus, we have the point ( 3 2 , 0 ) . So, if x = a = 3 2 , y = 0 .

Construct a coordinate system, plot these two points, and draw a line through them. Keep in mind that all the solutions to the equation 2 x + 3 y = 3 are precisely on this line.

A graph of a line passing through two points with coordinates zero, one and negative three over two, zero.

Got questions? Get instant answers now!

4 x + y = 5

To find the y -intercept , let x = 0 and y = b .

4 ( 0 ) + b = 5 0 + b = 5 b = 5

Thus, we have the point ( 0 , 5 ) . So, if x = 0 , y = b = 5 .

To find the x -intercept , let y = 0 and x = a .

4 a + 0 = 5 4 a = 5 a = 5 4

Thus, we have the point ( 5 4 , 0 ) . So, if x = a = 5 4 , y = 0 .

Construct a coordinate system, plot these two points, and draw a line through them.

A graph of a line passing through two points with coordinates zero, five and five over four, zero.

Got questions? Get instant answers now!

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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