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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.

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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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 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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