# Linear equations

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This chapter covers principles of linear equations. After completing this chapter students should be able to: graph a linear equation; find the slope of a line; determine an equation of a line; solve linear systems; and complete application problems using linear equations.

## Chapter overview

In this chapter, you will learn to:

1. Graph a linear equation.
2. Find the slope of a line.
3. Determine an equation of a line.
4. Solve linear systems.
5. Do application problems using linear equations.

## Graphing a linear equation

Equations whose graphs are straight lines are called linear equations . The following are some examples of linear equations:

$2x-3y=6$ , $3x=4y-7$ , $y=2x-5$ , $2y=3$ , and $x-2=0$ .

A line is completely determined by two points, therefore, to graph a linear equation, we need to find the coordinates of two points. This can be accomplished by choosing an arbitrary value for $x$ or $y$ and then solving for the other variable.

Graph the line: $y=3x+2$

We need to find the coordinates of at least two points.

We arbitrarily choose $x=-1$ , $x=0$ , and $x=1$ .

If $x=-1$ , then $y=3\left(-1\right)+2$ or $-1$ . Therefore, (–1, –1) is a point on this line.

If $x=0$ , then $y=3\left(0\right)+2$ or $y=2$ . Hence the point (0, 2).

If $x=1$ , then $y=5$ , and we get the point (1, 5). Below, the results are summarized, and the line is graphed.

 $X$ -1 0 1 $Y$ -1 2 5

Graph the line: $2x+y=4$

Again, we need to find coordinates of at least two points.

We arbitrarily choose $x=-1$ , $x=0$ and $y=2$ .

If $x=-1$ , then $2\left(-1\right)+y=4$ which results in $y=6$ . Therefore, (–1, 6) is a point on this line.

If $x=0$ , then $2\left(0\right)+y=4$ , which results in $y=4$ . Hence the point (0, 4).

If $y=2$ , then $2x+2=4$ , which yields $x=1$ , and gives the point (1, 2). The table below shows the points, and the line is graphed.

 $x$ -1 0 1 $y$ 6 4 2

The points at which a line crosses the coordinate axes are called the intercepts . When graphing a line, intercepts are preferred because they are easy to find. In order to find the x-intercept, we let $y=0$ , and to find the y-intercept, we let $x=0$ .

Find the intercepts of the line: $2x-3y=6$ , and graph.

To find the x-intercept, we let $y=0$ in our equation, and solve for $x$ .

$2x-3\left(0\right)=6$
$2x-0=6$
$2x=6$
$x=3$

Therefore, the x-intercept is 3.

Similarly by letting $x=0$ , we obtain the y-intercept which is -2.

If the x-intercept is 3, and the y-intercept is –2, then the corresponding points are (3, 0) and (0, –2), respectively.

In higher math, equations of lines are sometimes written in parametric form. For example, $x=3+2t$ , $y=1+t$ . The letter $t$ is called the parameter or the dummy variable. Parametric lines can be graphed by finding values for $x$ and $y$ by substituting numerical values for $t$ .

Graph the line given by the parametric equations: $x=3+2t$ , $y=1+t$

Let $t=0$ , 1 and 2, and then for each value of $t$ find the corresponding values for $x$ and $y$ .

The results are given in the table below.

 $t$ 0 1 2 $x$ 3 5 7 $y$ 1 2 3

## Horizontal and vertical lines

When an equation of a line has only one variable, the resulting graph is a horizontal or a vertical line.

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

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
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
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?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
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