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.
^z
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.
^z
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.
^z
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 .
^z
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
^z
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.
^z
   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


^z
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
^z
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

^z
          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                   }
^z
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
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
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
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
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.
^z
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.
^z
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
^z
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.
^z
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
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
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
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
-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
^z
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
^z
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.
^z
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.
^z
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 }

^z
            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.
^z
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 multiplying the top equation by blank

^t
Giving us
                  2x  + 6y = 20
                  4x  + 6y = 28

Next we will blank     the equations.

^z
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
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
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
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
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
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
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
^z







