<< Chapter < Page Chapter >> Page >

Find the equation of the line passing through ( −5 , 2 ) and ( 2 , 2 ) .

Horizontal line: y = 2

Got questions? Get instant answers now!

Determining whether graphs of lines are parallel or perpendicular

Parallel lines have the same slope and different y- intercepts. Lines that are parallel to each other will never intersect. For example, [link] shows the graphs of various lines with the same slope, m = 2.

Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7.  Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.
Parallel lines

All of the lines shown in the graph are parallel because they have the same slope and different y- intercepts.

Lines that are perpendicular intersect to form a 90° -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is −1 : m 1 m 2 = −1. For example, [link] shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of 1 3 .

m 1 m 2 = −1 3 ( 1 3 ) = −1
Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5.  Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x/3 minus 2.  Their intersection is marked by a box to show that it is a right angle.
Perpendicular lines

Graphing two equations, and determining whether the lines are parallel, perpendicular, or neither

Graph the equations of the given lines, and state whether they are parallel, perpendicular, or neither: 3 y = 4 x + 3 and 3 x 4 y = 8.

The first thing we want to do is rewrite the equations so that both equations are in slope-intercept form.

First equation:

3 y = −4 x + 3 y = 4 3 x + 1

Second equation:

3 x 4 y = 8 −4 y = −3 x + 8 y = 3 4 x - 2

See the graph of both lines in [link]

Coordinate plane with the x-axis ranging from negative 4 to 5 and the y-axis ranging from negative 4 to 4.  Two functions are graphed on the same plot: y = negative 4 times x/3 plus 1 and y = 3 times x/4 minus 2.  A box is placed at the intersection to note that it forms a right angle.

From the graph, we can see that the lines appear perpendicular, but we must compare the slopes.

m 1 = 4 3 m 2 = 3 4 m 1 m 2 = ( 4 3 ) ( 3 4 ) = −1

The slopes are negative reciprocals of each other, confirming that the lines are perpendicular.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph the two lines and determine whether they are parallel, perpendicular, or neither: 2 y x = 10 and 2 y = x + 4.

Parallel lines: equations are written in slope-intercept form.

Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 1 to 6.  Two functions are graphed on the same plot: y = x/2 plus 5 and y = x/2 plus 2.  The lines do not cross.
Got questions? Get instant answers now!

Writing the equations of lines parallel or perpendicular to a given line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line.

Given an equation for a line, write the equation of a line parallel or perpendicular to it.

  1. Find the slope of the given line. The easiest way to do this is to write the equation in slope-intercept form.
  2. Use the slope and the given point with the point-slope formula.
  3. Simplify the line to slope-intercept form and compare the equation to the given line.

Writing the equation of a line parallel to a given line passing through a given point

Write the equation of line parallel to a 5 x + 3 y = 1 and passing through the point ( 3 , 5 ) .

First, we will write the equation in slope-intercept form to find the slope.

5 x + 3 y = 1 3 y = 5 x + 1 y = 5 3 x + 1 3

The slope is m = 5 3 . The y- intercept is 1 3 , but that really does not enter into our problem, as the only thing we need for two lines to be parallel is the same slope. The one exception is that if the y- intercepts are the same, then the two lines are the same line. The next step is to use this slope and the given point with the point-slope formula.

y 5 = 5 3 ( x 3 ) y 5 = 5 3 x + 5 y = 5 3 x + 10

The equation of the line is y = 5 3 x + 10. See [link] .

Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 2 to 12 in intervals of 2.  Two functions are graphed on the same plot: y = negative 5 times x/3 plus 1/3 and y = negative 5 times x/3 plus 10.  The lines do not cross.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
Atega Reply
introduction to econometrics
Husseini Reply
Hi
mostafa
hi
LEMLEM
hello
Sammol
hi
Mahesh
bi
Ruqayat
hi
Ruqayat
Hi fellas
Nyawa
hey
Sammol
hi
God
hello
Jahara
Good morning
Jorge
hi
abubakar
hi
Nmesoma
hi
Mahesh
Hi
Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
explain inflationcause and itis degre
Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
Mercy Reply
Hi
Isiaka
Hi
Geli
hy
Man
👋
Bahunda
hy how are you?
Man
ys
Amisha
how are you guys
Sekou
f9 guys
Amisha
how are you guys
Sekou
ys am also fine
Amisha
fine and you guys
Geli
from Nepal
Amisha
nawalparasi district from belatari
Amisha
nd u
Amisha
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask