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An equation is a statement of
equality.
When we write, A = B, we mean that A
is equal to B. Since these letters represent numbers, we
are saying, the magnitude of A is equal to the magnitude of B.
Example:
25 = 25
true enough,
25 does equal 25. But, what is more important, is the
operations we can perform to the equation, maintaining
equality at all times, allowing us to "solve" for an
unknown.
In this example A is 25, and
B is 25. (Note: A = B can be written B = A)
Addition:
say we add
10 to A. Well, 25 + 10 is 35. To maintain
equality, we must also add 10 to B.
This equation becomes:
35 = 35
In other
words, if we add 10 to A, we must ALSO add 10 to B, if we do
not, our equation becomes 35 = 25, which we know
is incorrect, we would say 35 does not equal 25 and we
would write this fact this way: 35 ≠ 25.
Adding 10 to our equation ( A = B ) is shown:
A + 10 = B + 10
Adding the same quantity to
both sides of the equation maintains equality.
Subtraction:
suppose we subtract
5 from the equation 25 = 25. We'd have 25 - 5 which
is 20. To maintain equality, we must subtract 5 from
both sides of the equation.
The equation
then becomes: 20 = 20
Subtracting 5 from our equation ( A = B) is
shown:
A - 5 = B -
5
Subtracting the same
quantity to both sides of the equation maintains equality.
Multiplication:
Let's multiple by 3. 25 * 3 is 75.
The equation
25 = 25 then becomes: 75 = 75
This operation
to our equation A = B is shown:
3 * A = 3 * B
Multiplying the same
quantity to both sides of the equation maintains equality.
Side note: A student once
remarked: "I can solve all my problems, just multiply
the equation by 0 and all my problems vanish!" True
enough.
Division:
First of all, division
by 0 is undefined, therefore we will not allow division by 0.
We will allow division by all numbers except 0. (If we
divide by a variable, we must also state that at that step,
that variable cannot be 0.)
So, let's
divide by 5. 25 divided by 5 is 5.
The equation
25 = 25 becomes 5 = 5
This operation
to the equation A = B is shown:
A
/ 5 = B / 5
Dividing the same
non-zero quantity to both sides of the equation maintains
equality.
Note: many attempts using algebra
to prove one number equals another number of different
magnitude are done by dividing by the unknown and NOT
considering the case when that unknown is 0.
So, IN GENERAL, whatever mathematical
operation you do to one side of an equation, you must also do
to the other side.
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