Download equivalent forms

~ EQUIVALENT FORMS ~
Critical to understanding mathematics is the concept of equivalent forms.
Equivalent forms are used throughout this course.
Throughout mathematics one encounters equivalent forms of numbers,
equivalent forms of expressions, equivalent forms of equations, equivalent forms of
functions, equivalent forms of matrices, and the list goes on.
Examples of equivalent forms are given in this chapter,
Key Point: If two forms are equivalent, this indicated by an equals sign =, and
the forms are said to be 'equivalent'.
EQUIVALENT 1 THE MEANING OF EQUIVALENT NUMBERS
To say that two numbers are equivalent means that they live at the same place on
the number line.
EQUIVALENT 1.1
EXAMPLES OF EQUIVALENT NUMBERS
Ex 1: 5=10/2. That is, five is equivalent to ten divided by 2.
Why are 5 & 10/2 equivalent? Answer: Because both 5 and 10/2 live at the same
spot on the number line. Sqrt(25) also is equivalent to 5 for the same reason.
At this point you may say “gee there are lots of numbers equivalent to 5, such as
37-32, and 5.0 and 5/1 etc etc. If you think this, you are right! More importantly,
you understand the concept of equivalence of numbers.
Here are two more examples. Make sure you understand why the numbers in each
example are equivalent.
Ex 2: 17/17 = 1. Train yourself to think: “17/17 is equivalent to 1 because 1 lives at
the same spot on the number line as 17/17.
Ex 3: 2/Sqrt(2) = Sqrt(2). Train yourself to think: “2/Sqrt(2) is equivalent to
Sqrt(2) because 2/Sqrt(2) lives at the same spot on the number line as Sqrt(2).
Make sure you can show this mathematically.
EQUIVALENT 2 THE MEANING OF EQUIVALENT EXPRESSIONS
Two expressions are equivalent if they have the same value for any fixed value of
the variable. This is one tough concept for many students, so make sure you
understand it. Here are some examples:
EQUIVALENT 2.1
EXAMPLES OF EQUIVALENT EXPRESSIONS
Ex. a2+5a+6 = (a+3)(a+2). That is, a2+5a+6 is equivalent to (a+3)(a+2). Why are
a2+5a+6 and (a+3)(a+2) equivalent? Answer: Because, if you choose (“fix”) a to be
some number, say 7, then the value of a2+5a+6 is 90 which is the same as the value
of (a+3)(a+2). Check it out! Let a be, say, 13. In this case, the value of both
expressions a2+5a+6 and (a+3)(a+2) is 240. Check it out! This works, no matter what
values are chosen for a! That is what it means for two expressions to be equivalent!
Letters often used in mathematics are “x” & “y”, so you could state: “The
expression x2+5x+6 is equivalent to the expression (x+3)(x+2)
Here is another example, this time with two variables:
Ex. x2+2xy+ y2 = (x+y)2. That is, x2+2xy+ b2 is equivalent to (x+y)2. Why are x2+2xy+
y2 and (x+y)2 equivalent? Answer: Because, if you choose (“fix”) x to be some
number, say 3, and y to be another number, say 7, then the value of x2+2xy+ y2 is
100 which is the same as the value of the expression (x+y)2. Check it out! Also,
this is true if you let x be, say, 13, and y be, say, 29. In this case, the values of
both expressions x2+2xy+ y2 and (x+y)2 are 1764. Check it out! This works, no
matter what values are chosen for x & y! That is what it means for two expressions
to be equivalent!
Homework:
1. Prove that each of the following pairs of expressions are equivalent by factoring
the first to get the second.
2. Prove that each of the following pairs of expressions are equivalent by
multiplying the second to get the first.
3. For each pair of equivalent expressions, demonstrate the meaning of equivalence
as was done in Sec .2. That is, select random values for x & y, then use those values
to evaluate each expression in the pair, and observe that the numerical results are
the same.
(x3 + 23) & (x + 2)(x2 - 2x + 22)
(x3 - y3) & (x + y)(x2 + xy + y2)
(x2 - y2) & (x - y)(x + y)
(x2 - 1) & (x - 1)(x + 1)
(x - 11)2 & (x2 - 22x +121)
(x - y) (x - y) & (x2 - 2xy +y2)
x(x + 7) & x2 + 7x
x2 - 3x - 10 & (x - 5)(x + 2)
Note in this last example it is often easier to evaluate the factored form for
various values of x.
EQUIVALENT 3
THE MEANING OF EQUIVALENT EQUATIONS
Equivalent equations have the same solutions.
EQUIVALENT 3.1
EXAMPLES OF EQUIVALENT EQUATIONS
Here is an example of equivalent equations:
(x-2)(x+3) = 0
x2 + x - 6 = 0
x2 + x = 6
x2 = 6 - x
Each equation has the solutions 2 & -3, so they are all equivalent.
Homework:
Method of solution #1: Try to rewrite the original equation into an equation that is
equivalent to the choices. This is the more elegant way to solve this problem
because it requires mathematical skill rather than just brute force 'plug and chug'
as suggested next.
Method of solution #2: First solve the original equation, and then find which
equations have the same solutions.
1. Which of the following equations are equivalent to x2 - x - 6 = 0?
a) (x-2)(x+3) = 0
b) x2 - x + 6 = 0
c) x2 + x = 6
d) x2 = 6 + x
e) (x + 2)(x - 3) = 0
f) x2 - x = 6
g) x2 = 6 - x
h) 6 + x = x2
2. Which of the following equations are equivalent to 3x2 - x - 6 = 0?
a) (3x - 2)(x - 1) = 0
b) 3x2 - x = 6
c) 3x2 = x - 6
d) x2 = 6 + x
e) (x + 1)(3x - 2) = 0
f) x2 + x = 6
g) (3x + 2)(x - 1) = 0
h) 6 + x = 3x2
EQUIVALENT 4
Two functions
Two functions
Two functions
Two functions
are
are
are
are
THE MEANING OF EQUIVALENT FUNCTIONS
equivalent if they have the same graph.
equivalent if they have the same table.
equivalent if they can be written in the same form.
equivalent if they have the same verbal description.
EQUIVALENT 4.1
EXAMPLES OF EQUIVALENT FUNCTIONS
2
f(x) = x 5x + 6 is equivalent to the function g(x) = (x + 2)(x + 3).
In this case, it may not be immediately obvious that the function f is equal
to the function g. However, by factoring f, you get g. Like wise, by multiplying the
factors of g and collecting like terms, you get the function f. In either case you
can conclude that f = g.
Another way to show that f(x) = g(x) is to graph both functions. When you
do this, you will see that they both have the same graph.
You could also make tables for f & g and fill in several values in each table
until you are convinced that the ordered pairs in the tables are always the same.
This approach is not a proof as the algebraic approach above, but it can be very
convincing, which is often good enough.
EQUIVALENT 5 WIERD STUFF: EXPRESSIONS THAT ARE EQUIVALENT TO
NUMBERS
Sometimes an expression containing letters (variables) is equivalent to a
number. Amazing, but true! Consider the following examples:
Ex 1. (x-y)/(y-x) = -1. This says that (x-y)/(y-x) is equivalent to -1. Why are (xy)/(y-x) and -1 equivalent? Answer: Because, if you choose (“fix”) x to be some
number, say 7, and y to be another number, say 5, then the value of (x-y)/(y-x) is 1, which is the same as the value of the right hand side expression, -1. Check it out!
Also, this is true if you let x be, say, 13, and y be, say, 29. In this case, the values
of both expressions (x-y)/(y-x) and -1 are -1. Check it out! This works, no matter
what values are chosen for x & y! (except of course if x=y). That is what it means
for two expressions to be equivalent!
Ex 2. (a + b)/(a + b) = 1. Why is (a + b)/(a + b) is equivalent to 1?
EQUIVALENT 6 SUBTLE POINT: FINDING equivalent forms v.s. THE MEANING
OF equivalent forms.
There is a difference between what it means for two forms to be equivalent
and how to find those forms. That is, the meaning of equivalence is different than
the method of finding equivalent expressions.
For example, equivalent expressions can be found by factoring, expanding, or
multiplying by some form of 1 (see next chapter). But regardless of how the
equivalent expression is found, once you have two equivalent expressions, the
meaning of equivalence is that both expressions take on the same values for
specific values of the variables.
For example, a(a+b) can be shown to be equivalent to the expression (a2 + ab)
by using the distributive property. After this we have two equivalent expressions
a(a+b) & (a2 + ab). Now we can demonstrate the equivalence of these two
expressions by picking numbers for a & b, then use those numbers to evaluate each
of the two expressions and then observe that the two answers are the same! The
purpose of doing this second step is to gain an insight into, and understanding of
what it means to be equivalent.