In any language a  statement

conveys  a  fact  or  facts.

     A statement is a mathe-

matical  sentence which  may

be either true  or false.

eg:

   Four minus  three  equals

one       4 - 3 = 1

is a true statement ~ but

          4 - 2 = 1

is a false statement.

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An  open  sentence   on  the

other hand is one which  may

or may not  be  true.

eg:

   x + 4 = 7

This sentence is  true  only

if x = 3.

     An equation  is an open

sentence   with  an  unknown

quantity ~in the  above case

x~ which  can  be  found  by

solving the equation.

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Solving an equation involves

finding a value ( or values )

for the unknown quantity  so

that the  equation is true .

    The basic rule is to put

the unknown  quantity ( usu-

ally x )  on  the left  hand

side ( LHS ) of the equation

~ and the  known  quantities

on the ( RHS ) or visa versa.

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eg:       x  =  5 + 6

         LHS     RHS

          x  =  11

Two equations eg:

         x + 5 = 10

         and

         x + 6 = 11

are called equivalent equat-

ions because they both  have

the same value ( 5 ) for the

unknown quantity  x .

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^q

facts

always

may

open

unknown

equivalent

^t

A statement contains blank



^t

A Statement is blank  true or false.



^t

An open sentence blank be true or

false



^t

An equation is an blank sentence

^t

which contains an blank   quantity.



^t

Two equations having the same value

for their unknown quantity are

called blank      equations

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There  are four  basic rules for

simplify equations so  that they

can be solved.



1: Adding  the  same  number to

   both  sides  of an  equation

   doesn't change it.

   eg:

      x - 6     = 24

      x - 6 + 6 = 24 + 6

      x         = 30



2: Subtracting the same number

   from both sides of an equation

   doesn't change it.

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   eg:

     x + 8     = 11

     x + 8 - 8 = 11 - 8

     x         = 3



3: Multiplying both sides of an

   equation by the  same number

   doesn't change it.

   eg:

     +x + 3     =   7+

   2(+x + 3)    = 2(7+)

      x + 6     =  15

      x         =   9





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4: Dividing both sides of an

   equation by the same number

   doesn't change it.

   eg:

     2x + 8    =   18

   +(2x + 8)   = +(18)

      x + 4    =    9

      x        =    5

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Now  let's  put  these  rules

into  practice . Solve   this

equation.



      3x + 5 = 4(1 + x)

    Here's how . . . .



      3x + 5 = 4 + 4x

{ multiplying the terms

  inside the bracket by 4 }



  3x + 5 - 5 = 4 - 5 + 4x



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          3x = -1 + 4x

{ adding -5 to both sides

  of the equation         }



     3x - 4x = -1 + 4x - 4x

         -1x = -1

{ subtracting 4x from both

  sides                   }



           x = 1

{ multiplying both sides

  by -1                   }

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^q

2x

6x=3+12

6x=15

6

2+

^t

Solve these equations



4x = 3 - 2x + 12



add blank to both sides to eliminate

the x terms from the right hand side

^t

giving you blank



^t

which equals blank



^t

dividing both sides by blank



^t

gives you the value of x which is

blank

^z

^q

Add

8x-4=7x-2

adding -7x

x-4=-2

4

2

^t

3x - 4 + 5x = 7x -2



blank together the x terms on the

left hand side giving you

^t

blank



^t

Eliminate the x terms from the right

hand side by blank      to both

sides.

^t

giving you blank



^t

Add blank to both sides to find what

x equals.

^t

x = blank

^z

^q

Add

Subtract

-9=-5x

-5

9/5

^t

7x - 4 - 5 = 2x



blank together the numbers on the

left side of the equation.



^t

blank    7x from both sides



^t

giving you blank



^t

Divide -9 by blank to find what x

equals



^t

x = blank.

^z

^q

Subtracting

4y=7y-15

Subtracting

-3y=-15

Dividing

5

^t

  4y + 3 = 7y - 12



blank       3 from both sides of the

equation gives

^t

blank



^t



blank       7y from both sides gives

^t

us blank



^t

blank    both sides by -3 gives us

the value of y

^t

y = blank.

^z

^q

Multiply

-4y+3y=60-4y

Adding

3y=60

Dividing

20

^t

  -y + 3/4y = 15 - y



blank    both sides by 4 to get

rid of fractions

^t

Giving us blank



^t

blank  4y to both sides gives

^t

us blank



^t

blank    both sides by 3 gives us the

value of y

^t

y = blank.

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All this  is very fine ~ but

what  relevance  has  it  to

everyday life ~ what use has

it ?

    Well~try this  question.

John  is three times  Mary's

age but in  three years time

he  will  be twice her  age.

How  old are  John and Mary ?



    The answers are  John is

nine and Mary is three.

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How can you solve this equa-

tion mathematically ?

    John's age  is  given in

terms of Mary's  age .There-

fore let  Mary's  age be the

unknown quantity x.

    If you can  find  Mary's

age  x you  can  find John's

age 3x. At the present  time

    John = 3x years old.

    Mary =  x years old.

    In three years time

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John's present age + 3 years

= twice Mary's present age +

3 years .

    3x + 3 = 2(x + 3).

    3x + 3 = 2x + 6

    3x     = 2x + 3

         x = 3

Mary's  age   x = 3  years

John's  age  3x = 9  years

at the present time.

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How about trying another one.

The  last  one was difficult

and you  may not have under-

stood all the steps involved.

    This   time the  program

will  take  you through  the

steps  involved  and ask you

to input  the  right numbers

or words at each step.

^z

^q

x

equation

4

^t

Twelve plus a quarter of a  certain

number equals ~ seven plus half the

same number :  What is the number ?



We will represent the  number by

blank .



^t

12 + ,x = 7 + +x is the blank    of

the problem.



^t

To get rid of the fractions we will

multiply both sides by blank.



The  equation is  now  of  the form

48 + x = 28 + 2x .

^z

^q

28

x

^t

It can be simplified by subtracting

a number blank from both sides.



^t

48 - 28 + x = 28 -28 + 2x

     20 + x = 2x



Finally by subtracting blank from

both sides we get   x = 20.



Therefore the unknown number must

have equaled 20.

^z

^q

2x = 6x - 18

^t

Let's try another one.



Twice a certain number equals six

times the same number minus 18 ~

what is the number ?



First  the  equation ~ it's  the

mathematical way so don't try to

cheat by guessing.



The equation for the problem is

blank       .



^z

^q

-4x=-18

x=4+

4

^t

The next thing is to rewrite the

equation in a simpler form

blank     .



^t

Which can be rewritten as 4x = 18

simply by multiplying both sides by

-1.



This can in turn be simplified to

blank  .



^t

By dividing both sides by blank.



Giving you the number x = 4+

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So far all the  equations we

have  seen have had only one

unknown   quantity   usually

called x. Equations can have

as  many unknown  quantities

as you like.

    A linear equation is one

with two unknowns or variab-

les. ( x and y )

eg:

   x + y = 5

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They are so  called  because

they represent  the equation

of a straight line.

   Two linear equations  eg:

2x + y = 5   ~    x + 7y = 9

are called a set of :

Simultaneous Linear Equations

    It is possible given two

such  equations to find  the

values of x and y which will

make both equations true.

^z

   2x + y = 5  ~ x + 7y = 9

The only  values for  x an y

which  satisfy both  of  the

above  equations  are  x = 2

and y = 1.

    These values are  called

the solution set of the pair

of  simultaneous   equations

and are written as (2~1).

    You  can  check this  by

putting in 2 for x and 1 for

y in the above equations.

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How  did we  arrive at  this

pair of values for x and y ?

    There  are  two  methods

for  doing  this a graphical

method   and  an   algebraic

method.

    We  will concentrate  on

the algebraic  method  here.

    The   algebraic   method

uses the same techniques  we

used in solving equations of

one unknown.

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Solve algebraically the values of

x and y given that . . .



           2x +  y =  5

            x + 7y =  9



Rules:



  1:  Decide  which  variable you

      wish  to  solve for  first.



     { we will solve for y first }



^z



  2: Rewrite  the   equations  so

     that the  coefficient of the

     other  variable is the  same

     for both equations.



  { the coefficient is the number

    in front of the variable     }



  2x +  y =  5  -->  2x +  y =  5

   x + 7y =  9  -->  2x +14y = 18



{ multiplying the  bottom eqn by 2 }



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            2x +  y =  5

            2x +14y = 18





  3: Subtract the second equation

     from the  first  equation to

     eliminate the other variable



     { in this case we eliminate x

       so that we can solve y on its

       own                          }



^z

            2x +  y =  5

     -      2x +14y = 18

          --------------

               -13y =-13



  4: Find  the value  of the

     variable.



          {    y =  1    }



  5: Substitute the known variable

     into one of the equations.



     2x +  y =  5  -> 2x + 1 = 5

^z

  6: Solve for the unknown

     variable.



        { In this case x }



         2x = 4    x = 2



The solution set can be written as

               (x~y)

{ In this case (2~1) }



You  can check this  by substituting

the  values  for  x and y  into  the

equation.

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^q

y

2

subtract

^t

Try this example   x + 3y = 10

                  4x + 6y = 28



We will solve for x first.

To do this we must eliminate blank



^t

by 'x' the top equation by blank



^t

Giving us ........

                  2x  + 6y = 20

                  4x  + 6y = 28



Next we will blank     the equations.



^z

^q

4

substitute

2

^t

Giving us        -2x       =- 8

Therefore          x       =  blank



^t

We blank       this value for x into

one of the equations to find the

value of y.



^t

eg: 4x + 6y = 28 -> 16 + 6y = 28

Therefore          y       =  blank

^z

^q

y

adding

16x = 48

16

3

^t

Solve for x and y

                  4x - y =  0

                 12x + y = 48



We will solve for x first ~therefore

we must eliminate blank



^t

We do this by blank  both equations



^t

Giving us        blank



^t

Dividing both sides by blank

^t

gives us the value of x = blank

^z

^q

Substituting

12-y=0

12

(3~12)

^t

blank        this value of x into

the first equation gives us



^t

blank



^t

Therefore y = blank



^t

The solution set can be written as

blank

^z

^q

adding

-x=-2

2

Substituting

2-y=-1

^t

Solve for x and y

                 -2x + y = -1

                   x - y = -1



Solving for x first we must

eliminate y

We do this by blank  both equations



^t

Giving us        blank



^t

Therefore x = blank



^t

blank        this value of x into

the second equation gives us

^t

blank

^z

^q

2

-y=-3

3

(2~3)

^t

Subtracting blank from both sides

gives us

^t

blank



^t

Therefore y = blank



^t

And the solution set can be written

as blank

^z

^q

2x+y=1

-4x+3y=28

multiplying

adding

5y=30

^t

Solve      2x = 1 - y

           3y = 4x + 28



First we rewrite the equations

as         blank           and

^t

           blank



^t

Now lets solve for y first so we

eliminate x by blank       the

top equation by 2 and

^t

blank   both equations together.



^t

Giving us blank



^z

^q

6

2x=-5

2+

(-2+~6)

^t

Therefore y = blank



^t

Substituting this value into the

first equation we get blank



^t

Therefore x = blank

^t

the solution set can be written as

blank

^z

The Co - Ordinate plane is a

mathematical  drawing  sheet

which can show visually  the

equations which we have used

to solve problems and can in

itself  be  used as a quick

method of solving equations.

Remember the number line ?



<--------------------------->

 -6-5-4-3-2-1 0 1 2 3 4 5 6



^z

All  positive  and  negative

numbers  can  be represented

as points on the number line.

    Thus you have a means of

measuring  horizontal  dist-

ance . Consider  a  vertical

numberline.

         ^

         |  2

         |  1

         |  0

         | -1

         v

^z

Distance  in   the  vertical

irection  is measured  by a

vertical number line.

    By combining the  two it

is possible to measure dist-

ances in  both  the vertical

and horizontal directions.

    The  horizontal  axis is

called the x-axis.

    The   vertical  axis  is

called the y-axis.

^z

              ^    y-axis

              | 4

              | 3

              | 2

              | 1 2 3 4

<---------------------------->

     -4-3-2-1 |        x-axis

           -2 |

           -3 |

           -4 |

              v
















