# 11.3 Graphing with intercepts  (Page 2/4)

 Page 2 / 4

## Find the intercepts from an equation of a line

Recognizing that the $x\text{-intercept}$ occurs when $y$ is zero and that the $y\text{-intercept}$ occurs when $x$ is zero gives us a method to find the intercepts of a line from its equation. To find the $x\text{-intercept,}$ let $y=0$ and solve for $x.$ To find the $y\text{-intercept},$ let $x=0$ and solve for $y.$

## Find the x And y From the equation of a line

Use the equation to find:

• the x- intercept of the line, let $y=0$ and solve for x .
• the y- intercept of the line, let $x=0$ and solve for y .

x y
0
0

Find the intercepts of $2x+y=6$

To find the x- intercept, let $y=0$ :

 Substitute 0 for y . Add. Divide by 2. The x -intercept is (3, 0).

To find the y- intercept, let $x=0$ :

 Substitute 0 for x . Multiply. Add. The y -intercept is (0, 6).

The intercepts are the points $\left(3,0\right)$ and $\left(0,6\right)$ as shown in the chart.

Find the intercepts: $3x+y=12$

$\left(4,0\right)$ and $\left(0,12\right)$

Find the intercepts: $x+4y=8$

$\left(8,0\right)$ and $\left(0,2\right)$

Find the intercepts of $4x-3y=12.$

## Solution

To find the $x\text{-intercept,}$ let $y=0.$

 $4x-3y=12$ Distribute 0 for $y.$ $4x-3·0=12$ Multiply. $4x-0=12$ Subtract. $4x=12$ Divide by 4. $x=3$

The $x\text{-intercept}$ is $\left(3,0\right).$

To find the $y\text{-intercept},$ let $x=0.$

 $4x-3y=12$ Substitute 0 for $x.$ $4·0-3y=12$ Multiply. $0-3y=12$ Simplify. $-3y=12$ Divide by −3. $y=-4$

The $y\text{-intercept}$ is $\left(0,-4\right).$

The intercepts are the points $\left(-3,0\right)$ and $\left(0,-4\right).$

 $4x-3y=12$ x y $3$ $0$ $0$ $-4$

Find the intercepts of the line: $3x-4y=12.$

x -intercept (4,0); y -intercept: (0,−3)

Find the intercepts of the line: $2x-4y=8.$

x -intercept (4,0); y -intercept: (0,−2)

## Graph a line using the intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

Graph $-x+2y=6$ using intercepts.

## Solution

First, find the $x\text{-intercept}.$ Let $y=0,$

$\begin{array}{}\\ \phantom{\rule{0.7em}{0ex}}-x+2y=6\\ -x+2\left(0\right)=6\\ \phantom{\rule{2.8em}{0ex}}-x=6\\ \phantom{\rule{4.3em}{0ex}}x=-6\end{array}$

The $x\text{-intercept}$ is $\left(–6,0\right).$

Now find the $y\text{-intercept}.$ Let $x=0.$

$\begin{array}{}\\ -x+2y=6\\ -0+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=6\\ \phantom{\rule{3em}{0ex}}y=3\end{array}$

The $y\text{-intercept}$ is $\left(0,3\right).$

Find a third point. We’ll use $x=2,$

$\begin{array}{}\\ -x+2y=6\\ -2+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=8\\ \phantom{\rule{3em}{0ex}}y=4\end{array}$

A third solution to the equation is $\left(2,4\right).$

Summarize the three points in a table and then plot them on a graph.

 $-x+2y=6$ x y (x,y) $2$ $4$ $\left(2,4\right)$ $0$ $3$ $\left(0,3\right)$ $2$ $4$ $\left(2,4\right)$

Do the points line up? Yes, so draw line through the points.

Graph the line using the intercepts: $x-2y=4.$

Graph the line using the intercepts: $-x+3y=6.$

## Graph a line using the intercepts.

1. Find the $x-$ and $\text{y-intercepts}$ of the line.
• Let $y=0$ and solve for $x$
• Let $x=0$ and solve for $y.$
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.

Graph $4x-3y=12$ using intercepts.

## Solution

Find the intercepts and a third point.

We list the points and show the graph.

$4x-3y=12$
$x$ $y$ $\left(x,y\right)$
$3$ $0$ $\left(3,0\right)$
$0$ $-4$ $\left(0,-4\right)$
$6$ $4$ $\left(6,4\right)$

Graph the line using the intercepts: $5x-2y=10.$

Graph the line using the intercepts: $3x-4y=12.$

Graph $y=5x$ using the intercepts.

## Solution

This line has only one intercept! It is the point $\left(0,0\right).$

To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for $x,$ so we’ll let $x$ be $1$ and $-1.$

Organize the points in a table.

$y=5x$
$x$ $y$ $\left(x,y\right)$
$0$ $0$ $\left(0,0\right)$
$1$ $5$ $\left(1,5\right)$
$-1$ $-5$ $\left(-1,-5\right)$

Plot the three points, check that they line up, and draw the line.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Privacy Information Security Software Version 1.1a
Good
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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?