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 Solving 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 : Solving 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 Solving 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 : 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.
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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.
Progressive Learning of Math : Solving Equations
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