For
grade 6
Basic
Algebra.
Introduction:
It
is one of the basic branch of mathematics.
Just as the basic of Arithmetic are
the numbers 1, 2, 3, 4, ….for the
basic of algebra we use letters x, y,
z…For example if we give x the
value 2, y the value 3 and z the value
4. Then we can write 2, 3, 4 as x, y,
z. We can use algebra to find various
unknown values.
Like the ordinary numbers we can do
addition, multiplication, subtraction,
division in algebra. We can even use
brackets in it.
If
p & q are two numbers in algebra
then;
1. p + q means we are adding them.
2. p - q means we are subtracting y
from x.
3. p Í q means that they are
being multiplied.
4. p ¸ q means that q is being
divided by p.
5. p + ( p - q ) means that we first
subtract q from p and then add p in
it.
Algebraic
Variables:
x,
y, z ..etc are called algebraic variables
because their value "varies"
or changed.
For example,
x can take the value of 2 in one case
and it can take the value of 10 in the
second case.
These algebriac variables can also be
used with ordinary numbers.
For example,
1.
We multiply 5 with x as 5x.
2. We
can add 7 in y as y + 7.
3. We
can subtract z from 6 as 6 - z. and
so on…
Coefficient:
When
we use ordinary numbers with variables,
then these numbers are called coefficients.
For example,
In -5x, +74y, 69a, -21b etc, -5, +74,
69, & -21 are the coefficients.
Variables with no coefficient have the
coefficient 1 as in "x" or
"-y" the coefficients are
1 & -1 respectively.
Base
and exponents:
If
we multiply x three times we get,
x.x.x = x3
Here "x" is called the base
of the expression and "3"
is called the power or exponent. We
can always be added or subtracted. two
terms if they have same bases.
So 2x4 + 3x4 = 5x4.
Algebraic
Expression:
Combination
s of algebraic variables and coefficients
are called algebraic expressions.
For example,
3x, 2x + (5y - 3), 2y - 8 are all examples
of algebraic expression.
Evaluating
the Algebraic Expressions:
Each
variable in an algebraic expression
has a specific value and according to
these values the expression has a value.
This value is called "value of
expression".
For example, evaluating following,
1.
3x + 4 , when x = 2
3.2 + 4
6 + 4
10
The value of the given expression is
10.
2.
5x2 + 3x when x = 3
5*32+ 3.3
5*9 + 9
45 + 9
54
The value of the expression is 54.
3.
5x + ( 3y + 4 ) when x = 2 & y =
3
5*2 + ( 3*3 + 4)
10 + ( 9 + 4 )
10 + 13
23
The value of the expression is 23.
Algebraic
Equation:
When
we write 3x + 2 it does not mean anything
but if we write 3x + 2 = 10 it gives
us meaning and forms an algebraic sentence.
Relations:
Every
algebraic sentence represents a relation.
These relations are following:
Equal to
Not equal to
Greater than
Less than
Not greater than
Not less than.
Examples:
x + 5 = 10
x + 3 < 12
x + 7 > 2
Types
of sentences:
Depending
on these relations and values of the
variables in the relations there are
three types of sentences;
True
sentences.
3 + 4 > 2 is a true sentences because
3 + 4 = 7 & 7 > 2.
False
sentences.
6 + 5 = 10 is a false sentence because
6 + 5 = 11 & 11 is not equal to
10.
Open
sentences.
x
+ 3 = 7, is an open sentence because
it is neither true or false as we have
no information about x's value. This
open sentence has further two most frequently
used types, in algebra. They are;
·
Equations.
·
Inequations.
Equations:
This is a special kind of open sentence
in which we use the "equal to"
( = ) symbol. This open sentence can
be true or false depending on the of
value/s of the variable/s used. An equation
can have more than one variables.
Inequations:
They are commonly called the inequalities.
They use the signs ">"
or "<". They can also be
true or false depending on the of value/s
of the variable/s used.
Examples:
1.
Find if the following equations are
true of false.
x + 2 = 4 , if x = 4
2 + 2 = 4
4 = 4
The equation is true.
2.
Find the value of variable involved.
x
+ 4 = 10
x = 10 - 4
x = 6
The value of x is 6.
4
/ 5 - x = 1 / 5
- x = 1 / 5 - 4 /5
- x = - 3 / 5
x = 3 / 5
The value of x is 3 / 5.
Some
points to keep in mind while we solve
the equations:
1.
If one quantity is being added ( or
subtracted )on one side it will be subtracted
( or added )after we shift it to the
other side.
Example:
x - 5 = 10
x = 10 + 5
"5" was subtracted at the
right side and after shifting it was
added.
x
+ 10 = 15
x = 15 - 10
"10" was added at the right
side it was subtracted after shifting.
2.
If
one quantity is being multiplied ( or
divided ) at one side it will be divided
(or multiplied )to the other side.
Example:
y / 2 = 3
y = 2 x 3
y = 6
"2" was divided at one side
and when it was shifted it was multiplied.
3y
= 9
y = 9 / 3
y = 3
"3" was multiplied at one
side and after shifting to the other
side it was divided.
3.
If
both sides have minus signs they can
be canceled.
Example:
-x = -3
x = 3
Word
problems:
These
equations are used in many word problems.
Example:
If Jack has x sweets and Jane has twice
the number of sweets. What is the number
of sweets they have if the total number
of sweets is 12.
It is clear that the sum of sweets both
have is 12,
So ,
x + 2x = 12
3x = 12
x = 12 / 3
x = 4
Jack has 4 sweets.
And Jane has 2(4) = 8 sweets.
