SAMPLE LESSON 5
Joseph Buczek - Instructor
LESSON 5 - Equations, Identities and Inverses
This lesson explains the reasoning behind equations, identities and inverses of algebra
Equations, Identities and Inverses
An equation is a mathematical statement of equality. This means that an equation represents, or is a picture of an equality between two real-life facts. An equals sign (=) separates the statements of equality.
WHAT IS AN EQUATION?
A problem immediately faces us when we look to algebra as a method of determining an unknown value of something that is mixed, or combined, with a bunch of other, known things. The problem is: How would we set up our algebra system to find the unknown value in terms of the, other, known values?
Well, the ancient mathematicians must have struggled a lot to find the answer. It took a lot of years - perhaps thousands - before they finally figured the way to do it. Now that we know how they did it, it seems so easy. But, it took millions of years before man discovered the wheel and, of course, that seems to us so easy to discover, too. When you are dealing with a strict and accurate thing like calculating something with numbers you can't use techniques that are subject to error under certain conditions. If there are conditions under which the mathematics you are using is not going to be correct you want to know about these.
The ancient mathematicians knew that if they were to devise a system of finding an unknown value blended, or mixed, with a bunch of known values they had to:
A) somehow devise a method by which they could separate the unknown value from the mixture of known values (i.e., get the unknown on one side and the known values on the other side of a line.
B) find a sure method of calculating, or determining, the correct value of the unknown quantity in terms of the other, known values.
The solution to this problem was arrived at when philosophers thought of using A STATEMENT OF EQUALITY between two quantities, or expressions. This statement of equality created by them was called an EQUATION. As will be seen, both of the above conditions are fulfilled by the use of the equation. An equation allows a person to solve for an unknown value of a quantity when the unknown is combined, or mixed, with several known quantities.
Understanding, now, that the use of the concept of the equation is the unique, or single, method by which an unknown value may be found, and, understanding that finding unknown values is very important to people today because it is the tool which helps people to predict future events, it becomes apparent that learning about equations is an important part of our educational process.
Now that we appreciate that an equation, or mathematical statement of equality, is the device, or tool, used to solve for unknown values hidden within an expression, we need to look into some real equations to see how we can use them. The principal part of an equation is the equal sign (=). This sign is placed between two separate mathematical expressions that (through consideration of the conditions in the problem) are known to have the same, identical mathematical values. For example, consider the following:
PROBLEM 1: When 14 is added to a number the sum is 36. What is the number?
SOLUTION: Let us call the unknown number x. From the statement of the problem we can determine that
x + 14 = 36
which signifies that the value of the expression x + 14 is identical to the value of the number 36. We have expressed our problem in terms of a statement of equality. We have expressed our problem in terms of an equation .
Now that we have an equation expressing the conditions of our problem how do we find the value of the unknown number, which we call x? As described in A), above, we should first separate our unknown value, x, from the rest of the expressions, or values, in the problem. How do we do this? In this case we use the arithmetic process of subtraction to eliminate the +14 from the left side of the equation. One of the fundamental procedures that is known about equations is that you can add, subtract, multiply or divide both sides of an equation by a number without altering the equality of the equation. So, we add -14 to both sides of the equation (this is the same as subtracting +14 from both sides). Doing this results in
x + 14 - 14 = 36 - 14
14 - 14 = 0, so the above equation is transformed to
x = 36 - 14
But, 36 - 14 = 22, so
x = 22
We have found the solution to the problem through calculation. The number, x, which, when added to 14 produces the number 36 is x = 22. Through the use of the equality statement, or equation, we have solved the problem.
PROBLEM 2: Twenty four less than a certain number is 33. Find the unknown number.
SOLUTION: Using an equality statement in the form of an equation we arrange the above as
x - 24 = 33
The first thing we need to do to solve this equation for x is to get this equation into a form whereby the unknown value, x, is on one side of the equation and all the other terms are on the other side (it doesn't matter which side of the equation x is on). We need to eliminate the -24 from the left side of the equation. We do this by adding + 24 to both sides of the equation, as
x - 24 + 24 = 33 + 24
-24 + 24 = 0, so the above equation can be rearranged to
x = 33 + 24 = 57
So, the number, which, when 24 is subtracted from, results in 33 is 57. Through the use of the statement of mathematical equality, or equation, we have solved the problem.
PROBLEM 3: Fifty six is 8 times a certain number, find the number.
SOLUTION: The problem here involves multiplication. We may arrange the problem in the form of a mathematical equality, or equation, as
8x = 56
If we are to solve this equation for x we must isolate x from all the other quantities within the equation. The only quantity intermingled with the x is the 8. Since we may divide both sides of an equation by the same number without altering the equality of the equation, we eliminate the 8 along side the x by dividing both sides of the equation by 8, as,
(8/8)x `=` 56/8
8/8 = 1 so the above equation can be changed to
x = 56/8 = 7
PROBLEM 4: Nine less than 26 is eighty-five divided by a certain number. Find the number.
SOLUTION: The problem can be expressed in a mathematical statement, or equation form, as
26 - 9 =`85/x
Since we need to isolate x on one side of the equation we must eliminate the 85 from the right side of the equation. We do this by dividing both sides of the equation by 85. Notice that the subtraction of the 25 - 9 expression may be performed before the division, or the entire expression may be divided by 85, as
(26- 9}/85`=`(85/x) (1/85)
26 - 9 = 17 and (85)(1/85) = 1, so,
This equation is ok, except that the unknown value, x, that we need to find, is in the denominator (lower part of the fraction) on the right side of the equation. To get the x into the numerator (top side of a fraction) we can multiply both sides of the equation by x. Let us do that below,
x(17)/85 = (1/x)(x)
Now, (1/x)(x) = x/x = 1 so,
(17/85)x =` 1
Now, we seem to have gotten ourselves into a complicated condition with the x connected to the 17/85. This is what happens when you solve this kind of problem. The solution is easy: move the 17/85 over to the right side. This is achieved by multiplying the 1/85 on the left side of the equation by 85 and multiplying the 17 on the left side by 1/17. But, what you do to one side of an equation, you must do to the other side. The result is
(85/17)(17/85)x = 85/17
(85/17)(17/85) = 1 so,
x = 85/17`= 5
So, through the use of the mathematical statement of equality, or equation, we have determined the solution to the problem nine less than twenty six is eighty five divided by a certain number.
Once you know the techinique of finding this solution, the equaion 17/85(x) = 1 may be quickly solved by merely inverting the 17/85 to 85/17 and placing it on the right side. Many students who already know this method and have solved several problems using it may suddenly do this and confuse a student who doesn't know what they are doing. But, once you work with a few equations and accustom yourself to this, you will do this too, in order to save time. If you are uncertain, however, always follow the procedure shown and you will always arrive at a correct answer if you don't put in an error of you own during the calculations.
In conclusion, we see that knowing how to set up and solve a problem in the form of a mathematical statement, or equation opens a students' learning experience and enhances his/her capability to have mastery over math problems.
The word identity refers to one's sense of being separated, or distinguishable from other person. Identity refers to individuality. It signifies being who or what one is. In mathematics, an equation can have significance in what it is, too. In mathematics an identity is a statement of equality that is true for all values of a variable. For example,
6x + 3x = 9x
is a mathematical statement of equality, or an equation, that is true for all values of x. If x = 3, then, substituting this value into the above equation results in
6(3) + 3(3) = 9(3)
18 + 9 = 27
Substituting x = 22 into the original equation results in
6(22) + 3(22) = 9(22)
Doing the multiplications results in
132 + 66 = 198
or, 198 = 198
The object is that an identity is a statement of equality that is true for all values of a variable (the x in the original equation). The equation 6x + 3x = 9x is true for all values of x. This is not true for all equations. Some equations are true for only one value of x. Because the equation 6x + 3x = 9x is true for all values of x, this equation is called an identity. Because the expression on the left side of the equation reduces to the same expression that is present on the right side of the equation, (i.e., 6x + 3x equals 9x) the equation is called an identity. This is because every permissible real value that x can have is a solution of the equation. An equation that is true for all values of the variable, x, indicates this identity condition by the use of three dashes as in
6x + 3x = 9x
instead of the usual two dashes used for an equals sign (=). The identity, which holds for all values of a variable, is different from an equation, which holds for only one value of an equation. An equation that holds for only one value of a variable, x, is called a conditional equation. A conditional equation is an equation like
9x = 27
here, the only value that x can have is x = 3 since 9(3) = 27 and cannot equal anything else.
An identity element is a number that, when added to, or multiplied with another number, results in that identical number. The number zero is the identity element for addition. This can be shown by adding
7 + 0 = 7
+1026 + 0 = 1026
0 + 2bx = 2bx
The number 1 is the identity element for multiplication because, for example,
(25)(1) = 25
(6k2x)(1) = 6k2x
THE INVERSE OF A NUMBER
The word inverse designates a condition in which something is reversed in sequence, or order. In algebra the term is used to indicate numbers that are reversed. The additive inverse of a positive number is a negative number. The additive inverse of a positive number is a negative number. The additive inverse of 3 is - 3. The additive inverse of -26 is +26.
Consider the fraction 1/17. The multiplicative inverse of a number is the number 1 divided by that number. For example, the multiplicative inverse of 17 is 1/17. So, 1/17 is also called the multiplicative inverse or reciprocal of 17. The multiplicative inverse of 77 is 1 over 77. The number 1/77 is called the reciprocal or 77. The reciprocal of 1/77 is 77.
Algebraically, the inverse properties are:
x + (-x) = (-x) + x = 0
x(1/x)= (1/x)x = x/x = 1