SOLVING EQUATIONS - INTRODUCTION TO SOLVING, DOMAIN, EXAMPLES, LINKS FOR FURTHER STUDY

Please study

Math Equations before Solving Equations,

if you have not already done so.

There we discussed about Mathematical Sentence,
Mathematical Statement, Equality, Open Sentence,
Equation, Solution or Root of an Equation.

That knowledge is a prerequisite here.

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.

Domain of the variable : Math Equations

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:

1. 3x - 5 = 9
2. 2x + 3y = 7
3. 3x + y + 2z = 25
   
4. p³ = 27
5. 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 : Solving Equations

An equation in which the highest exponent of the varaibles presentis one is called a Linear Equation.

Equations (i), (ii), (iii) given above are Linear Equations.

For the Solutions of Linear Equations, go to
Linear Equations.

and for Word Problems on Linear Equations go to
Algebra Word Problems.

For the Solutions of Linear Equations with
two variables including Word Problems, go to
Linear Equations in two variables.

Quadratic Equations : Solving Equations

An equation in which the highest exponent of the varaibles presentis two is called a Quadratic Equation.

Equation (v) given above is Quadratic Equation.

For the Solutions of Quadratic Equations by Factoring go to

Solving Quadratic Equations by Factoring.

For the Solutions of Quadratic Equations by Formula, go to

Quadratic Formula.

and for Word Problems on Quadratic Equations go to

Math Word Problems.

Cubic, Bi-Quadratic etc Equations : Solving 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.