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Algebra for Adults:
Part 12, Rules of Equality:
February 24, 2003:
First, the problems from last 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?
Let x represent Orville's hourly rate.
Orville's gross pay will be equal to his rate times his hours.
So 72x = $792.00
x = $792.00/72 = $11.00
Orville's hourly rate was $11.00 per hour.
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?
Let y represent William's initial weight.
His initial weight will be equal to his current weight less his
weight gain.
y = 320 lb. - 132 lb.
y = 188 lb.
William used to weigh 188 lbs.
Chuck's 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?
Let a represent the capacity of the gas tank.
Miles per gallon (MPG) will be equal to total miles driven divided
by the capacity of the tank.
19 MPG = 437 miles/ a gallons.
a = 437/19
a = 23 gallons.
The tanks capacity is 23 gallons.
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?
Let l represent the yards lost.
The yards lost will be the difference between the first down yard
line and the fourth down yard line.
l = 27 - 6 = 21 yards.
Detroit lost 21 yard on their three and out drive.
Maybe next year.
Now we can look at a few useful rules.
These will be very handy tools for solving some problems.
The Addition Rule of Equality:
If equal numbers are added to both sides of an equation the equality
does not change.
Let's try that and see.
Take a simple example.
Z - 50 = 100
Now let's try the addition rule of equality.
By rule, we can add 50 to both sides of the equation.
Z - 50 + 50 = 100 + 50
Now we can simplify this thing.
The -50 and + 50 will cancel.
So Z = 100 + 50 = 150
We can check our solution by plugging it into the original statement.
150 - 50 = 100.
Sur enough, it checks.
Now, since addition and subtraction are inverse operations,
can you guess that there is a similar rule for subtraction?
We would be shocked if there were not because algebra is, above
all else, logical.
The Subtraction Rule of Equality:
If equal numbers are added to both sides of an equation the equality
does not change.
To prove this, we can take a modified version of our simple equation.
Z + 50 = 150
Now we can try the subtraction rule of equality.
By rule, we can subtract 50 from both sides.
Z + 50 - 50 = 150 - 50
Now we can simplify.
The 50 and - 50 will cancel.
So Z = 150 - 50 = 100
I'll leave it to you to prove this by substituting the solution into the original equation.
Okay for addition and subtraction.
I bet you have already guessed that we have similar rules for multiplication and division.
You bet.
Here they are.
Multiplication Rule of Equality:
If both sides of an equation are multiplied by the same number,
the equality does not change.
Division Rule of Equality:
If both sides of an equation are divided by the same number, the
value does not change.
Let's try it for multiplication with a simple equation.
Z/16 = 2
Let's multiply both sides of this thing by 16 and resolve it.
Z x 16/16 = 2 x 16
Recall from basic math that the 16/16 will cancel.
So Z = 2 x 16 = 32
The proof is simple enough. We plug 32 into the original equation.
32/16 = 2
Now prove the division rule of equality to yourself with this equation.
B x 20 = 100
Can you see that these two rules are just a different way of saying a rule we learned 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.
Now how can we summarize these four rules in a single statement
to make it simpler for us?
How about this?
We do not change the equality of an equation when we perform
the same operation on both sides using the same number except
division by zero.
For sure, we know that division by zero is impossible or undefined.
Now apply these rules to solve the following problems.
We will check the answers next time.
15z = 45
6a + 16 = 34
c - 35 = 42
c + 2c - 27 = 3
7y - 2y = 35
Eddie worked 8 hours and earned $120.00.
What was his hourly rate?
Lydia was an advertising actress. She made TV commercials.
One 30 second spot she made was very successful. It ran 19,320
times before it was mothballed. Lydia got paid one percent of
her union scale for each showing of her film. Her union scale
was $235.00 per hour, or per appearance, whichever pays more.
How much did Lydia earn on that film?
Back to Algebra for Adults.
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