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Adults.
Algebra for Adults:
Part 10, Simple Equations:
February 10, 2003:
It is not necessary to be intimidated by the notion of an equation.
Remember, we defined an equation way back in mathematics.
That definition is still valid.
We will notice it again and again. In Mathematics, the rules and
definitions never change. A rule is a rule forever. A definition
will always be true.
So an equation is still,
A statement asserting the equality of two expressions, usually
written as a linear array of symbols that are separated into left
and right sides and joined by an equal sign.
Now let's play with that.
Here is a simple definition of an equality.
An equality is a statement that two expressions have the same
value.
That means, the expressions are equal to each other.
For example:
5x = 10 is an equality.
2y + 6 = 18 is an equality.
3z + z = 4z is an equality.
10 = 10 is an equality.
Thus an equality is a lot like an equation.
So far so good.
Now for algebra we will want to get more specific about an
equation.
We will find out why shortly.
So, here is a narrower definition,
An equation is an equality wherein the unknown quantity or quantities may have only one value or values.
So, the example 5x = 10 must be an equation since x can only
be equal to 2.
Remember how we solved that?
5x = 10 is the same as x = 10/5, thus x = 2.
Okay, we can see that.
Notice that we did not invalidate our earlier definition.
It is still true, but it is very general compared to our algebra
definition.
This narrower definition allows us to define something slightly different called an identity.
An identity is a equality in which a letter or letters may have any value.
For example:
The statement 4y + 5y = 9y is an identity since y may have any
value.
Let's work it out.
Substitute 3 for y.
The statement becomes (4 x 3) + (5 x 3) = (9 x 3)
So 12 + 15 = 27 or 27 = 27
Follow this through on your own. You will find no matter what value you substitute for y the statement is still true.
So, we see an identity is very much like an equation.
Here is the only difference.
An equation is a conditional equality. Only one value of the unknown
will satisfy it.
An identity is an unconditional equality. The unknown may have
any value.
Cool!
Now let's define the root of an equation.
A root of an equation is the number which when substituted for
the unknown, will make the equation equal.
So the root is just the solution to the equation.
For example:
In the statement 5z + 4 = 19 the root is 3.
3 is the only value that will satisfy the equation.
Work it out.
5z + 4 = 19, or (5 x 3) + 4 = 19, or 15 + 4 = 19.
Try any other number and you will find that 3 is the only value of z that will satisfy the equation.
So far, so good.
Now, here are a few exercises for you.
We can check your answers next time.
Identify each of these as an equation or an identity.
2x + 7 = 21
5z - 5 = 20
2a + 3a = 5a
10 - 2b = 6
14 = 3c + 5
8y = 2 y + 2 y + 4y
Find the root of 17c = 34
Find the root of 35x = 35
Back to Algebra for Adults.
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