<< Chapter < Page
  Linear equations   Page 1 / 1
Chapter >> Page >

Equations and inequalities: linear simultaneous equations

Thus far, all equations that have been encountered have one unknown variable that must be solved for. When two unknown variables need to be solved for, two equations are required and these equations are known as simultaneous equations. The solutions to the system of simultaneous equations are the values of the unknown variables which satisfy the system of equations simultaneously, that means at the same time. In general, if there are n unknown variables, then n equations are required to obtain a solution for each of the n variables.

An example of a system of simultaneous equations is:

2 x + 2 y = 1 2 - x 3 y + 1 = 2

Finding solutions

In order to find a numerical value for an unknown variable, one must have at least as many independent equations as variables. We solve simultaneous equations graphically and algebraically.

Khan academy video on simultaneous equations - 1

Graphical solution

Simultaneous equations can be solved graphically. If the graph corresponding to each equation is drawn, then the solution to the system of simultaneous equations is the co-ordinate of the point at which both graphs intersect.

x = 2 y y = 2 x - 3

Draw the graphs of the two equations in [link] .

The intersection of the two graphs is ( 2 , 1 ) . So the solution to the system of simultaneous equations in [link] is y = 1 and x = 2 .

This can be shown algebraically as:

x = 2 y ∴ y = 2 ( 2 y ) - 3 y - 4 y = - 3 - 3 y = - 3 y = 1 Substitute into the first equation: x = 2 ( 1 ) = 2

Solve the following system of simultaneous equations graphically.

4 y + 3 x = 100 4 y - 19 x = 12
  1. For the first equation:

    4 y + 3 x = 100 4 y = 100 - 3 x y = 25 - 3 4 x

    and for the second equation:

    4 y - 19 x = 12 4 y = 19 x + 12 y = 19 4 x + 3

  2. The graphs intersect at ( 4 , 22 ) .

  3. x = 4 y = 22

Solution by substitution

A common algebraic technique is the substitution method: try to solve one of the equations for one of the variables and substitute the result into the other equations, thereby reducing the number of equations and the number of variables by 1. Continue until you reach a single equation with a single variable, which (hopefully) can be solved; back substitution then allows checking the values for the other variables.

In the example [link] , we first solve the first equation for x :

x = 1 2 - y

and substitute this result into the second equation:

2 - x 3 y + 1 = 2 2 - ( 1 2 - y ) 3 y + 1 = 2 2 - ( 1 2 - y ) = 2 ( 3 y + 1 ) 2 - 1 2 + y = 6 y + 2 y - 6 y = - 2 + 1 2 + 2 - 5 y = 1 2 y = - 1 10
∴ x = 1 2 - y = 1 2 - ( - 1 10 ) = 6 10 = 3 5

The solution for the system of simultaneous equations [link] is:

x = 3 5 y = - 1 10

Solve the following system of simultaneous equations:

4 y + 3 x = 100 4 y - 19 x = 12
  1. If the question does not explicitly ask for a graphical solution, then the system of equations should be solved algebraically.
  2. 4 y + 3 x = 100 3 x = 100 - 4 y x = 100 - 4 y 3
  3. 4 y - 19 ( 100 - 4 y 3 ) = 12 12 y - 19 ( 100 - 4 y ) = 36 12 y - 1900 + 76 y = 36 88 y = 1936 y = 22
  4. x = 100 - 4 ( 22 ) 3 = 100 - 88 3 = 12 3 = 4
  5. 4 ( 22 ) + 3 ( 4 ) = 88 + 12 = 100 4 ( 22 ) - 19 ( 4 ) = 88 - 76 = 12

A shop sells bicycles and tricycles. In total there are 7 cycles (cycles includes both bicycles and tricycles) and 19 wheels. Determine how many of each there are, if a bicycle has two wheels and a tricycle has three wheels.

  1. The number of bicycles and the number of tricycles are required.

  2. If b is the number of bicycles and t is the number of tricycles, then:

    b + t = 7 2 b + 3 t = 19
  3. b = 7 - t Into second equation: 2 ( 7 - t ) + 3 t = 19 14 - 2 t + 3 t = 19 t = 5 Into first equation: : b = 7 - 5 = 2
  4. 2 + 5 = 7 2 ( 2 ) + 3 ( 5 ) = 4 + 15 = 19

Simultaneous equations

  1. Solve graphically and confirm your answer algebraically: 3 a - 2 b 7 = 0 , a - 4 b + 1 = 0
  2. Solve algebraically: 15 c + 11 d - 132 = 0 , 2 c + 3 d - 59 = 0
  3. Solve algebraically: - 18 e - 18 + 3 f = 0 , e - 4 f + 47 = 0
  4. Solve graphically: x + 2 y = 7 , x + y = 0

Questions & Answers

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 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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Linear equations. OpenStax CNX. Jun 15, 2015 Download for free at https://legacy.cnx.org/content/col11828/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Linear equations' conversation and receive update notifications?

Ask