MATH EQUATIONS - EQUATIONS, TYPES, SOLUTION OR ROOT(S), DOMAIN, LINKS TO OTHERS

Please study
Basic Algebra before Math Equations,
if you have not already done so.

There, we introduced the idea of literal number.

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.

Mathematical Statement : Math Equations

Some mathematical sentences are given below.

(i) 8 + 7 = 15. (ii) 9 + 4 > 14 (iii) 16 + 12 < 30 (iv) 3 + 7 ≠ 10

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.

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.

Equation : Math Equations

An Open Sentence containing the sign ' is equal to' ( = ) is called an Equation.

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.

Solving Equations, Domain of the Variable, Examples

For lucid explanation of the basics of
Solving Equations and the concept of
Domain of the variable with Examples,
and for Links to Solving Linear, Quadratic,
Cubic and Bi-Quadratic Equations, go to

Solving Equations