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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
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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
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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
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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

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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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there is no specific books for beginners but there is book called principle of nanotechnology
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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.
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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
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of graphene you mean?
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or in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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