Introductuon to Equations

An equation is an algebraic statement that two expressions are equal. An equation consists of:

*.a left-hand-side expression (abbreviated LHS)
*.a right-hand-side expression (abbreviated RHS), and
*.an equal sign between them.

Here are three examples:
1. 5 + 3 = 8
2. 5 + 3 = 9
3. x+ 4 = 10
The first example is an equation that is always true; 5 plus 3 always equals 8. This is also known as anidentity. The second example is an equation that is never true; 5 plus 3 never equals 9. This is also known as a contradiction.
The third example is known as a conditional equation. It is only true under the condition that the variable, x, equals 6 since only 6 plus 4 equals 10. No other value of x makes the equation true. In this book we are primarily interested in conditional equations.

Solutions

Finding all the values of the variable x that make the left-hand-side of an equation equal the right-hand-side is called solving the equation for x. When we set out to find these values, x is called the unknown. Once we have found the values they are called the solutions or roots of the equation. We say that they make the equation true. We also say that these values of x satisfy the equation.
Equations can have any number of solutions, as these examples show.

Example:
The equation:
has one solution, namely x= 3. We can verify that this solution satisfies the equation by substituting it back into the equation.
This gives:
Since both the left-hand-side and the right-hand-side simplify to 1/6, the solution x= 3 is verified.
Example:
The equation:
20x3+ 25x2+ 5x= 0 has three solutions: x= 0,x= −¼ and x= −1.
This is called a solution set. We can verify that all three values of x satisfy the equation by substituting each of them into the equation and getting the true statement that 0 = 0.

Example:
Some equations have no solutions over the real numbers but do have solutions over the complex numbers. An example is the equation:

x2= −9.

There is no real number which when squared will yield −9 but there are two complex numbers, namely 3i and −3i.

Example:
Some equations have no solutions whatsoever. They are called contradictions. An example is the equation:

x=x+ 1.
There is no number that is unchanged when you add 1 to it. Checking the solutions Once you have found a possible solution for an equation you must substitute it into the original equationand check that it does indeed satisfy the equation. There are two reasons: (1) you may have made an error in solving the equation, and (2) the algebra process that we will describe below for solving an equation can produce so-called “extraneous solutions” (essentially garbage solutions!) You can’t tell that you have an extraneous solution until you substitute the solution in and discover that it doesn’t satisfy the original equation. Extraneous solutions must be rejected.
In our next post we'll be talking on the TYPES OF EQUATIONS we have.....i'm sure you've learnt alot in today's post, you can tell us using the comment box