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Adults.
Algebra for Adults:
Part 11, Verbal Statements:
February 17, 2003:
Here are a the exercises from last time.
Identify each of these as an equation or an identity.
2x + 7 = 21, equation.
5z - 5 = 20, equation
2a + 3a = 5a, identity.
10 - 2b = 6, equation.
14 = 3c + 5, equation.
8y = 2 y + 2 y + 4y, identity.
Find the root of 17c = 34
c = 34/17 = 2, root = 2.
Find the root of 35x = 35
x = 35/35 = 1, root = 1.
On these two problems we applied a rule we learned back in
basic math.
We can move any term across an = sign without changing the value
of the equation by also moving the term across the division sign.
We will learn a different way of thinking about this later.
Onward and upward.
In the real world, problems do not usually come to us as neat
little equations ready for solution. Most often we get a bunch
of verbiage with a problem embedded in it.
For a simple example:
Six times what number equals thirty?
The first thing to do in these things is to assign a symbol
(letter) to the unknown.
In this case, we can let z represent the unknown number.
Then our statement becomes six times z = thirty or 6z = 30.
That is really all there is to it.
Let's try one that is a little more verbose.
Five times a number less five is equal to twenty. What is the number?
The same procedure still works.
Let y represent the unknown number.
Then five times y less five is equal to twenty.
So, 5y - 5 = 20.
To find the number we must solve this equation for y.
We will find out how later.
For now I will just lay it out.
5y - 5 = 20
5y = 20 + 5
y = 25/5 = 5
Okay.
Let's try one that is more like the real world we could wish for.
John bought a painting at a flea market. Later he sold it to his manager for five times what he paid. His profit was eighty dollars. How much did he pay for the painting?
First we can let z represent what he paid for the painting.
That is our unknown.
He sold it to his boss for five times that value or 5z.
His profit was eighty dollars.
Profit is the difference between what he paid for the painting
and what he sold it for.
That is 5z - z.
Now we can write out the equation and solve it.
5z - z = $80.00.
4z = $80.00
z = $80.00/4
z = $20.00
John paid twenty bucks for the painting.
Not bad!
Okay.
Now let's reiterate something we already know from regular math.
This is a rule.
Addition and subtraction are inverse operations.
Did we already know this?
Sure!
We used this fact to check our problems in arithmetic.
For example:
4 + 5 = 9 means that 9 - 5 = 4.
We used subtraction to check addition.
Now we can notice one additional thing about this.
If addition is involved in the unknown of an equation, we can
use subtraction to solve for the unknown.
For example:
In the statement, y + 6 = 10, addition is involve in the unknown.
By rule then, we use subtraction to solve the problem.
So, y = 10 - 6
y = 4
Piece of cake.
Now, what did we really do here?
We moved the plus six across the equal sign and changed it's sign.
That is the whole procedure.
Now we can get a little more general about it.
Inverse operations are two operations such that if one is involved with the unknown in an equation, the other is used to solve the equation.
Sure!
We just demonstrated that above with addition and subtraction.
Do you suppose the same thing applies to multiplication and division?
Darn right!
Multiplication and division are inverse operations.
This is the same thing we learned in regular math.
We can use multiplication to check division and vice versa.
Some things never change.
For example:
In the statement, 6a = 30, multiplication is involve with the
unknown.
Therefore, we use division to solve the equation.
a = 30/6 = 5
How bout that?
This simply agrees with our rule from above.
We can move any term across an = sign without changing the value
of the equation by also moving the term across the division sign.
Enough already!
Here are some teasers.
We'll check the answers next time.
Orville was a working fool. Last week he worked 72 hours. His gross pay for that time came to $792.00. What was Orville's hourly rate?
William was jilted by his true love and went on an eating binge. When he finally checked in at Food Fools Anonymous, he weighed in at an incredible 320 lbs. He had gained 132 lbs. How much did he weigh before he went on the binge?
Chucks car gets nineteen miles to the gallon of gas. He filled his car with gas and took a drive up north. When he ran out of gas, he had traveled 437 miles. How much gas does his tank hold?
Detroit received the kickoff and returned it to their own twenty-seven
yard line. They had the ball first down and ten yards to go. When
they punted four downs later they punted from their own six yard
line. How many yards did they lose before they punted?
Back to Algebra for Adults.
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