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Algebra for Adults:
Part 14, Second Review Solutions:
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In algebra 10 we defined an equality.
An equality is a statement that two expressions have the same
value.
That means, the expressions are equal to each other.
5x = 10 is an equality.
We defined an equation as: an equality wherein the unknown quantity or quantities may have only one value or values.
We also found an identity to be an equality in which a letter or letters may have any value.
For example:
The statement 4y + 5y = 9y is an identity since y may have any
value.
An equation is a conditional equality. Because only one value
of the unknown will satisfy it.
An identity is an unconditional equality because the unknown may
have any value.
We also defined the root of an equation as The number which when substituted for the unknown, will make the equation equal.
So the root is just the solution to the equation.
For example:
In the statement 5z + 4 = 19 the root is 3.
2x + 78 = 22, is an equation.
5z - 4 = 21, is an equation
2a + 3a = 5a, is an identity.
Find the root of 18c = 36
c = 36/18 = 2, root = 2.
Derive equations for these statements.
Six times what number equals thirty?
6z = 30.
Five times a number less ten is equal to fifteen.
5y - 10 = 15.
Addition and subtraction are inverse operations.
If addition is involved in the unknown of an equation, we can use subtraction to solve for the unknown.
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.
Multiplication and division are inverse operations.
We can use multiplication to check division and vice versa.
The Addition Rule of Equality is: If equal numbers are added to both sides of an equation the equality does not change.
The Subtraction Rule of Equality is: If equal numbers are added to both sides of an equation the equality does not change.
The Multiplication Rule of Equality is: If both sides of an equation are multiplied by the same number, the equality does not change.
The Division Rule of Equality is: If both sides of an equation are divided by the same number, the value does not change.
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.
Solve the following problems.
15z = 45
z = 45/15 = 3
6a + 16 = 34
6a = 34 - 16
6a = 18
a = 18/6 = 3
c - 35 = 42
c = 42 + 35 = 77
c + 2c - 27 = 3
3c - 27 = 3
3c = 3 + 27
c = 30/3 = 3
7y - 2y = 35
5y = 35
y = 35/5 = 7
Formally, a formula is a statement or equation of a fact, rule, or other logical relationship.
In other words, a formula is just a standardized equation for a particular kind of situation.
If we traveled in our auto for two and one half hours at a rate of sixty four miles per hour, how far would we have traveled?
The rate is 64 MPH.
The time is 2.5 hours.
By formula:
D = 60 x 3 = 180 miles.
This concludes the study of basic algebra. To be sure, there is much more to mathematics and even algebra for those who wish to pursue it. For those who do, the Schaum's Outlines are an outstanding source of lessons in advanced math. I hope you will continue. This is truly the stuff that spaceships are made of. The magic of mathematics is not just an interesting adventure. The people at NASA, the people at General Motors, the people at Disneyland, and the man who composed the 1812 overture all had to understand the basic principles of mathematics that underlie their disciplines before they could do what they did. As you continue your study you will discover these relationships between math and all other science.
William:
Back to Algebra for Adults.
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