Basic Algebra
Arithmetic with the Letter X
Basic Algebra is simply doing arithmetic with
at least one number replaced by a letter. This letter is referred to as
an unknown and can be either a variable (can have a variety of values)
or a constant (has only one value).
For example, you know how to do
3 + 2 = 5
This is an Arithmetic Equation. If we replace 2 with x we have
3 + x = 5
This is an Algebra Equation. We know from our actions and just by
looking at the equation that in this case
x = 2
Algebra 1 is simply a set of rules that find out the value of x.
Before we explain these simple rules we need to have a brief discussion
about what an equation is.
An equation is like a "Teeter-Totter" that is perfectly balanced with a
an equal weight on each end. This is illustrated in the picture
below.
Basic Arithmetic Equation
An arithmetic equation is made up of a series of numbers and
operators (+,-,*,/) on the right and left with an equal sign in between.
This equation is represented in the picture above as a perfectly
balanced Teeter-Totter with an equal total weight on each side. The
equation in our example is
3 + 2 = 5
By definition an the total value on the left side must be equal to the
total value on the right side. Otherwise the "Teeter-Totter" will not be
balanced.
There is one critically important property of an equation that this
picture is intended to illustrate. That point is: If you do exactly
the same thing to both sides of an equation then you still have an
equation - both sides will still be equal and the "Teeter-Totter" is
still balanced.
This is illustrated in the picture above. If you add 2 to both sides
of the equation as in the third Teeter-Totter, you can see that you have
7 lbs total on each side and the Teeter-totter is still balanced.
If you subtract 2 from each side as in the last Teeter-Totter, then you
have 3 lbs on each side and the Teeter-Totter is still balanced.
An Arithmetic Equation can be changed into an Algebra Equation by
replacing one of the numbers with X as shown in the picture below.
Basic Algebra Equation
When the 2 in the Arithmetic Equation 3 + 2 = 5 is replaced by X the
Algebra Equation 3 + X = 5 is obtained as shown by the first Teeter-
Totter in the above picture.
When 3 is subtracted to both sides of the algebra equation we have left
an X lb weight on the left side and a 2 lb weight on the right side and
the Teeter-Totter is still perfectly balanced.
This actually represents the solution to the equation, which is X = 2.
In the picture below we illustrate a second Basic Algebra Equation and show
you how to solve it.
Basic Algebra Equation #2a
In this example we use a different Basic Algebra Equation 2X + 3 = 13.
In the second Teeter-Totter we replace the 13lb weight with a 10 lb
weight and a 3 lb and the Teeter-Totter remains balanced.
The next step shown in Teeter-Totter number three is to subtract 3 from
each side which leaves us with 2X = 10 and the Teeter-Totter still
perfectly balanced.
This not yet a solution as we have 2X remaining on the left side instead
of just X. For a solution see the Teeter-Totters in the picture below.
Basic Algebra Equation #2b
Remember, if we do exactly the same thing to both sides of an equation
the result is still an equation - that is, the Teeter-Totter is still
balanced.
So, in the second Teeter-Totter above, we divide both sides by 2. When
we divide something by 2 it is the same thing as cutting it into two
halves. It we divide a 10 lb weight in half we get two 5 lb weights and
if we divide a 2X lb weight in half we get two X lb weights.
So the result of dividing both sides by 2 is the solution X = 5.
The steps necessary to solve any Basic Algebra equation are summarized in the picture below.
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