Appendix�C
Algebra
| C.1� |
Proportions and Equations |
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Physics deals with physical variables and the relations between them. Typically, variables are represented by the letters of the English and Greek alphabets. Sometimes, the relation between variables is expressed as a proportion or inverse proportion. Other times, however, it is more convenient or necessary to express the relation by means of an equation, which is governed by the rules of algebra.
If two variables are
directly proportional and one of them doubles, then the other variable also doubles. Similarly, if one variable is reduced to one-half its original value, then the other is also reduced to one-half its original value. In general, if
x is directly proportional to
y, then increasing or decreasing one variable by a given factor causes the other variable to change in the same way by the same factor. This kind of relation is expressed as
, where the symbol
means “is proportional to.”
Since the proportional variables
x and
y always increase and decrease by the same factor, the ratio of
x to
y must have a constant value, or
, where
k is a constant, independent of the values for
x and
y. Consequently, a proportionality such as
can also be expressed in the form of an equation:
. The constant
k is referred to as a
proportionality constant.
If two variables are
inversely proportional and one of them increases by a given factor, then the other decreases by the same factor. An inverse proportion is written as
. This kind of proportionality is equivalent to the following equation:
, where
k is a proportionality constant, independent of
x and
y.
| C.2� |
Solving Equations |
|
Some of the variables in an equation typically have known values, and some do not. It is often necessary to solve the equation so that a variable whose value is unknown is expressed in terms of the known quantities.
In the process of solving an equation, it is permissible to manipulate the equation in any way, as long as a change made on one side of the equals sign is also made on the other side. For example, consider the equation
. Suppose values for
v,
, and
a are available, and the value of
t is required. To solve the equation for
t, we begin by subtracting
from
both sides:
Next, we divide both sides of
by the quantity
a:
On the right side, the
a in the numerator divided by the
a in the denominator equals one, so that
It is always possible to check the correctness of the algebraic manipulations performed in solving an equation by substituting the answer back into the original equation. In the previous example, we substitute the answer for
t into
:
The result
implies that our algebraic manipulations were done correctly.
Algebraic manipulations other than addition, subtraction, multiplication, and division may play a role in solving an equation. The same basic rule applies, however: Whatever is done to the left side of an equation must also be done to the right side. As another example, suppose it is necessary to express
in terms of
v,
a, and
x, where
. By subtracting
from both sides, we isolate
on the right:
To solve for
, we take the positive and negative square root of
both sides of
:
| C.3� |
Simultaneous Equations |
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When more than one variable in a single equation is unknown, additional equations are needed if solutions are to be found for all of the unknown quantities. Thus, the equation
cannot be solved by itself to give unique values for both
x and
y. However, if
x and
y also (i.e., simultaneously) obey the equation
, then both unknowns can be found.
There are a number of methods by which such simultaneous equations can be solved. One method is to solve one equation for
x in terms of
y and substitute the result into the other equation to obtain an expression containing only the single unknown variable
y. The equation
, for instance, can be solved for
x by adding
to each side, with the result that
. The substitution of this expression for
x into the equation
is shown below:
We find, then, that
, a result that can be solved for
y:
Dividing both sides of this result by 11 shows that
. The value of
can be substituted in either of the original equations to obtain a value for
x:
| C.4� |
The Quadratic Formula |
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Equations occur in physics that include the square of a variable. Such equations are said to be
quadratic in that variable, and often can be put into the following form:
where
a,
b, and
c are constants independent of
x. This equation can be solved to give the
quadratic formula, which is
The
in the quadratic formula indicates that there are two solutions. For instance, if
, then
, and
. The quadratic formula gives the two solutions as follows:
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