We also explained the basic operations like addition,
subtraction, multiplication and division on literal numbers.
We also Explained about Algebraic Expression.
We saw with examples, how to find the value of
an Algebraic Expression by substituting the given
values to the variables involved.
That knowledge is a prerequisite here.
Here, we consider an Algebraic Expression with "=" symbol.
We also discuss the set of values, the variable(s)
can take (replacement set) and the values out of
the replacement set that make the algebraic expression
equal to the R.H.S. of the "=" symbol.
Equation : Math Equations
Before answering the question what is an equation ?
we need to know about certain terminology.
Mathematical Sentence : Math Equations
We have seen that the combination of terms, obtained by the
operation of '+' or '-' or both is called an Algebraic Expression.
If the terms are only numerical (without literals),
the expression is called numerical expression. e.g. 9 + 4, 6 - 3, 15 + 7 - 2, are numerical expressions.
If a numerical expression and a number or
two numerical expressions are joined or connected by
' is equal to' ( = ) or 'is greater than' ( > ) or 'is less than' ( < ) etc,
they are called Mathematical Sentences.
Each of the above sentences can be verified as either true or false.
You can see Sentences (i) and (iii) are true and
Sentences (ii) and (iv) are false.
A Mathematical Sentence that can be verified as either true or false
but not both is called a Mathematical Statement.
All the four examples given above are Mathematical Statements.
Equality : Math Equations
A true mathematical statement containing the sign ' is equal to' ( = )
is called an Equality.
In the above examples, example (i) is an equality.
Open Sentence : Math Equations
So far we have seen Numerical Expressions joined by
' is equal to' ( = ) or 'is greater than' ( > ) or 'is less than' ( < ) etc,
and called them Mathematical Sentences.
Now consider Sentences in which Algebraic Expressions
with literals (variables) connected by
' is equal to' ( = ) or 'is greater than' ( > ) or 'is less than' ( < ) etc.
What to call them?
The sentences containing variables depend upon the value of the
variable for their truth or falsity.
Such sentences may be true for particular
value(s) of the varaible and false for others.
Such sentences are known as open sentences.
Thus,
A sentence which contains a variable such that it may be true or false
depending on the values of the variable, is called an
Open Sentence.
Some Open Sentences are given below.
(i) x + 3 = 9. (ii) y + 2 > 17 (iii) z - 4 < 5 (iv) p + 8 ≠ 6
Can Movies be Learning Tools ?
Do you know Movies provide us
Entertainment, Social interaction,
Laughing Therapy, Learning Tool,
Relaxation, General Well Being ?
An Open Sentence containing the sign ' is equal to' ( = )
is called an Equation.
In the above examples, example (i) is an equation.
Note that every Equation has two sides, namely
'Left Hand Side' (L.H.S.) and 'Right Hand Side' (R.H.S.).
Thus, in the equation x + 3 = 9, L.H.S. is x + 3 and R.H.S. is 9.
Solution or Root of an Equation : Math Equations
A number which when replaced for the variable of an equation
makes the resulting statement true. i.e., makes its L.H.S. is equal to
its R.H.S. is said to satisfy the equation. A number which thus satisfies
an equation is called a Solution or a Root of the equation.
Example 1 of Math Equations
Consider the equation x + 3 = 9.
You can see that the number 6 when replaced for the variable x
makes the L.H.S. and the R.H.S. of the Equation equal.
Hence 6 is called a root or a solution of the equation x + 3 = 9.
Example 2 of Math Equations
Some times an Equation can have more than one root.
e.g. Consider the equation x2 = 4.
You know 2 x 2 = 22 = 4;
and also -2 x -2 = (-2)2 = 4.
Thus both 2 and -2 satisfy the equation x2 = 4.
∴The equation x2 = 4 has two roots 2 and -2.
{2,-2} is called the solution set of the equation x2 = 4.
To solve an equation means to find its Solution Set
or finding all the Roots of the equation.
We have seen, The solution of an equation is the number(s)
to be replaced for the variable such that L.H.S. = R.H.S.
The set of values from which we can replace the variable
is called Replacement set.
Look at the following Example.
Example 3 of Math Equations
Solve the equation x + 1 = 0,
(i) If the replacement set for the variable x is taken as
Whole number set i.e. W = { 0, 1, 2, 3,.......}.
(ii) If the replacement set for the variable x is taken as
Integer set i.e. Z = { ......-3, -2, -1, 0, 1, 2, 3,.......}.
Solution:
(i) There is no solution.
(ii) there is solution ( i.e. -1) for the equation.
The replacement set of the variable of an equation
is called the Domain of the variable.
Unless otherwise stated, the Domain of the variable
is taken as the Real Number Set.
Kinds of Equations : Math Equations
Consider the following equations:
3x - 5 = 9
2x + 3y = 7
3x + y + 2z = 25
p3 = 27
m2 + 5m + 6 = 0
From the above equations, we observe the following facts:
An Equation may contain more than one variable.
Equations (i), (iv), (v) have one variable.
(ii) has two variables x and y and
(iii) has three variables x, y, z.
The highest exponent or index of the variable
in an equation may be more than one.
In Equations (i), (ii), (iii), the highest exponent of the variable is 1,
in Equation (iv), the highest exponent of the variable is 3
and in Equation (v), the highest exponent of the variable is 2.
Linear Equations : Math Equations
An equation in which the highest exponent of the varaibles present
is one is called a Linear Equation.
Equations (i), (ii), (iii) given above are Linear Equations.
Cubic, Bi-Quadratic etc Equations : Math Equations
For solving Equations with the highest
exponent of the varaible present being more than two, i.e. Cubic Equations, Bi-Quadratic Equations etc.,
we need Theory of Equations which is covered in Algebra Equations.
