~ 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 THE MEANING OF EQUIVALENT FUNCTIONS Two functions are equivalent if they have the same graph.
Two functions are equivalent if they have the same table.
Two functions are equivalent if they can be written in the same form.
Two functions are equivalent if they have the same verbal description.
EQUIVALENT 4.1 EXAMPLES OF EQUIVALENT FUNCTIONS f(x) = x2 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 (x- y)/(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.
