|
CALCULUS I MATH 181 |
Southwestern Adventist University |
|
| Distance Education | Lawrence E. Turner, Jr., Ph.D. | |
course syllabus
assignments
materials
request a test
proctor form
A basic construct in algebra is the equation. It is a statement that has two parts connected by an equal sign, and states that the left hand expression has the same value as the right hand expression. As an example:
This statement is a True statement. Generally, when we write any equation, we make the unstated assertion that the statement or equation is true.
Typically we have an equation involving a single variable such as x. The original equation then is True for a given set of values of x. In the simplest linear case, the solution set is a single value. As an example:
2x − 3 = 4x + 1
The solution to this equation is x = −2. This can be verified by substituting this value for x in the original equation:
| left hand side: | 2(−2) − 3 = −7 | right hand side: | 4(−2) + 1 = −7 |
Indeed the left hand side (LHS) does have the same value as the right hand side (RHS) verifying that this value of x does produce a true statement.
The process of algebra involves manipulating the original equation so as to isolate all the x's on one side with a coefficient of one and all the remaining quantities on the other. This is done by a series of operations to both sides of the equation that retain its truth. As an example, 3 can be added to both sides of the equation above. The resulting equation is still true and is simpler than the original.
The primary problem with algebra is to know what such operation to do at each point and to perform the operations without error!
In set notation, the solution set above could be written as:
x ∈ {−2}
For quadratic equations, such as:
x2 + x − 6 = 0
the solution set may take on two distinct values. In this case, x = 2 and x = −3 are solutions as can be verified by substitution. The solution set is:
x ∈ {−3, 2}
However, there may be other possibilities. Consider:
x = |x|
The solution set in this case is a range of values:
Another, rather bizarre situation is an equation like:
To solve this subtract the fractions on the left side to simplify the left side to a simple fraction and then equate it to the right side:
For this to be true, the numerator on the left must equal the numerator on the right since the denominators are the same. (Another way is multiply both sides by (x−1).) Therefore:
.However, this creates an impossible situation. When substituted into the original equation, we get a division by zero. Actually, there was an unstated restriction in the original. It really should have been:
The solution obtained by ordinary algebra manipulations is the restricted value. Therefore, the equation has no solution! Now, an interesting question is whether the original is actually a true statement to begin with?
One more fundamental possibility occurs with an equation like:
The right side is simply the two factors of the left hand side. This is one of the possible algebraic manipulations (in this case substitution for one quantity with its equal). However, if we think of it as an equation to be solved, we can ask: what is the solution set? The answer is actually simple: x can take on any real (actually any complex) value. In symbolic form:
This is really good! The left hand side can be substituted for the right hand side at any point in the algebraic manipulation without affecting the possible results. Often this type of equation is called an identity and is occasionally written as:
where the "equal to" symbol has been replaced by the "identically equal to" symbol.
Typically written in a more general form:
where it is understood that x and a are not values to solve for, but are parameters that can take on any real value.
And finally, we have occasion to evaluate a quantity as a decimal number rather than an exact symbolic value. Consider the solution to:
The solution set contains two values:
or
The solutions written above are exact. The use of the "equal to" symbol is correct. However, sometimes it is desired to provide a decimal value and (for the positive value) write:
strictly speaking, this is not correct. It should be written as:
The "approximately equal to" symbol is correct. Since the exact value is an irrational number, no matter how many decimal places are written, the decimal expansion is still only an approximation. Now, not all decimal expansions are approximations. For
the decimal expression
is exact and correct since the fraction is a terminating decimal number.
Typically in the sciences where values are ultimately taken from measurements and are approximations, the "equal to" symbol is used throughout, but from a perfect mathematical perspective should generally be replaced by the" approximately equal to" symbol whenever actual approximate numbers are substituted for algebraic symbolic quantities.
© 1999, 2000, 2001, 2002, 2003, 2008 by Lawrence Turner