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Basics of Algebra
• Algebra operates on the idea that an equation represents a scale. Instead of keeping the
scale balanced with weights,
we use numbers, or constants.
These numbers are called constants
because they constantly have the same value.
• Next we use a device called a variable, or an unknown number represented by any letter
in the alphabet (usually x). The value of each variable must remain the same in each
problem.
• Finally, we use several symbols to relate all of these variables and constants together.
These symbols are listed and explained below.
Multiply, if x is used as variable then don’t use it as multiplication
· or * or x
sign
Divide
/ or 
+
Add or Positive
-
Subtract or Negative
Calculate what is inside of the parenthesis first. (also called grouping
symbols)
()
Basics of the Equation:
•
•
Solving Equations
Pretend you have a scale like the one shown. On the right side there are 45 pennies and
on the left side are 23 pennies and an unknown amount of pennies. The scale is balanced;
therefore, we know that there must be an equal amount of weight on each side.
The equation
x + 23 = 45
Subtract 23 from each side
- 23 -23
Result
x = 22
Equation with a constant and a variable on each side
The equation
x + 23 = 3x +45
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Basics of the Proportion
A proportion is a special form of an algebra equation. It is used to compare two ratios or make
equivalent fractions. A ratio is a comparison between two values. Such as the following:
1 apple: 3 oranges
This ratio compares apples to oranges. It means for every apple there are 3 oranges.
Proportions are two equivalent ratios A proportion will help you solve problems like the one
below.
Sara has a box of apples and oranges in the ratio of 2:3. If she has six apples, how many oranges
does she have?
Before we begin to set up proportions for a word problem, we will concentrate on solving
proportions. Remember, a proportion is a comparison between two ratios. The proportion shown
below compares two ratios which are in the fraction form.
2
6
=
3
x
•The four parts of the proportion are separated into two groups, the means and the extremes. The
extremes are the very first number, and the very last number. This can be remembered because
they are at the extreme beginning and the extreme end. The means are the second and third
numbers. This can be remembered because mean is another word for average or in the middle.
Both the means and the extremes are illustrated below.
Solving a Proportion
•Algebra properties tell us that the product of the means is equal to the product of the extremes.
You should know that the fraction one-half is equal to two-fourths. This is shown as a proportion
below.
1
2
=
2
4
Because they are equal, the product of the means is equal to the product of the extremes, this is
shown below.
2*2=1*4
4=4
•This property is extremely useful when one of the means or one of the extremes is unknown.
Now, we can use this property to solve the previous question.
2
6
=
3
x
3
•To solve this, and find the value of x: write an equation, on the left side multiply the means, on
the right side, multiply the extremes. Then solve the equation for x.
Solving a proportion with two variables:
•A proportion with two of the same variables can also be solved. Take the problem below for
example.
25 x
=
x
1
•When you encounter a situation like the above, a variable squared equals a number, you can do
one of two things.
•1. Find what number squared is equal to 25. Use our perfect squares chart for reference.
•(or) 2. Change the problem to x = the squareroot of 25. The resulting number from either
method will be equal to x and will be the answer.
When a whole number is in place of a fraction
Take a look at the problem below, notice it doesn't have a fraction on one side.
To solve this proportion, you have to change the whole number to a fraction, just as you did in
math class, by putting it over a 1. The problem above would turn into the following:
It could then be solved like any other proportion.
What is simplifying?
•In math class you simplified fractions to make them easier to understand and work with. For
example, the first fraction, as shown below, can be simplified to a much smaller fraction with the
same value.
E.g.
4
•Simplifying multiple add or subtract signs: This is the most basic method for simplifying
equations or expressions. If you were having a conversation with someone and said "I am not,
not hungry." it would actually mean that you were hungry because you used double negatives.
This can lead to confusion, therefore it is considered improper to use double negatives in speech.
•It is okay for an equation, like the first one below, to have more than one sign between two
terms. If there are two or more negative or subtract signs, like in the second one, it should be
simplified before it is solved.
•Example:
Combining like terms
•A term is a constant or a variable in an expression. In the equation
12+3x+2x^2=5x-1
•12, 3x, and 2x^2 are the terms on the left, and 5x, and -1 are the terms on the right.
To
combine like terms, you combine or add up all of the like terms.
•For example, each of the following are like terms because they are all x with a coefficient. 2x,
45x, x, 0x, -26x, -x
•Each of the following are like terms because they are all constants. 15, -2, 903, 0.6
•Each of the following are like terms because they are all y^2 with a coefficient. 3y^2, y^2, y^2, 26y^2
•For comparison, below are a few examples of unlike terms.
•The following two terms are both variables but because they are different letter variables, they
are unlike terms. 17x, 17z
•The following terms are all the same letter, but each letter has a different power or exponent.
15y, 19y^2, 31y^5
The following two terms both have the letter y but the second term has another variable in it,
therefore they are unlike terms. 19x, 14xy
Example 1: x + 3x + 7 = 42 + x - 12
To begin solving this equation, we will simplify the left hand side. The first step is to find which
terms can be added. The x and 3x are like terms, so they are added and become 4x. The 7 does
not have a like term, so it can be left alone. The equation should now read:
4x + 7 = 42 + x - 12
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The next step is to simplify the right hand side of the equation. This time there is no term which
can be added with x, but there are two constants: 42 and -12. When added they become 30. The
equation will now read like the one below.
4x + 7 = x + 30 The equation can now be solved as usual.
example 2
-9 + 12x - 10 - 4x = 8x - 6x + 46 - - 1
Simplifying Multiplication
First, the commutative property. This is an algebra property which tells us that the
product of a multiplication problem is the same, regardless of the order the terms were multiplied
in.
For example, 2 x 10 equals 20 and 10 x 2, also equals 20. This property is also true when
multiplying more than 2 terms, for example 4 x 6 x 5 equals 120, and 6 x 5 x 4 also equals 120.
The next thing you should know is how to show multiplication in algebra. Because x is used in
expressions and equations as a variable, it would be extremely confusing if it were also the
multiply sign. So instead of using an x we can show the multiplication several different ways.
Each of the examples on the right mean x squared times two.
Multiplication of Variables
The first problem we will be working on is shown above. When we multiply same letter
variables, we keep the letter and add exponents. Since neither x has a visible exponent, their
exponents are both 1. Since we are multiplying, we will add the exponents 1 + 1 and get 2. The
answer is shown below.
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Look over the next problem:
Simplifying Using Distribution
The distributive property is an algebra property which helps to multiply a single term and two or
more terms inside parenthesis. Take a look at the problem on the right.
2(3+6)
Because the 3 + 6 is in parenthesis, you must first find the answer of 3 + 6, then multiply it by 2.
This will give you an answer of 18. Read over the expression below.
6(2+4x)
The distributive property tells us that we can remove the parenthesis if the term that the
parenthesis is being multiplied by ( ) is distributed to, or multiplied with each term inside the
parenthesis. For example,
FOIL Method
The FOIL Method is a process used in algebra to multiply two parenthesis.
FOIL stands for:
An example of this is given below.
(3 + 7x)(6 + 2x) =
7
More than two terms in a parenthesis
Example: (x+3)(2x2 + 10x +1)
Exponents
Exponents are shorthand for multiplication. The exponents stand for however many times the
thing is being multiplied. The thing being multiplied is called the base. If a is any real number
and n is any positive integer, then by an we mean the quantity
a.a. . . . a (n times);
thus, a1 = a,
a2 = a.a,
a5 = a.a.a.a.a
In the expression an the number n is called the exponent, and the number a is called the base.
E.g. 55 is 5 raised to the fifth power and is equal to 5*5*5*5*5.
E.g. compute the following two examples
Exponent Identities
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Special Cases
The zero exponent
Each one of these problems is solved by using a 1 multiplied by the number, the amount of times
the exponent says to. If the exponent is 0, the one isn't multiplied by the number therefore, the
answer is 1.
This is an important rule to remember. It will be explained better in one
of the next sections of this lesson. E.g. 50 = 1
Zero with an exponent:
In most cases zero with an exponent can be calculated like any other number and exponent. The
exception is 00. Zero with an exponent of zero is undefined, and cannot be calculated.
Be careful not to confuse this rule with the last one! Zero to any power except 0 is always zero,
because no matter how many times you multiply the 1 by zero the answer will always be zero.
The 1 exponent:
Because anything to the first power is the same as if it had no exponent, the exponent can be
simply removed.
E.g. 51 = 5
Variables of Exponents
Exponents of variables do the same thing - tell you how many times it is multiplied. Take a look
at the example below.
Example:
Simplifying Negative Exponents
Negative exponents show how many times 1 is divided by a number.
E.g. write 5-2 using only positive exponents
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Polynomial Exponents
Because the two terms inside the parenthesis are not being multiplied or divided, the exponent
outside the parenthesis can't just be "distributed" in. Instead, you must be multiplied by the entire
parenthesis the amount of times the exponent tells you to. In this problem the exponent is 2, so it
is multiplied two times:
This method will also work when the terms are being divided, like the problem below:
Fractional (Rational) Exponents
We can use fractional exponents for expressions involving radicals as follows
Radical
Notation
Exponential Notation
Example
__
a
a1/2 (or a0.5)
641/2 = 8
3
__
a
a1/3
641/3 = 4
__
a
a1/n
321/5 = 2
n
In general, we can use the following rule:
n __
m
a
or
n __
a
a
m
m/n
323/5 = (321/5)3
= 23 = 8
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Order of Operations
•The Order of Operations is very important when simplifying expressions and equations. The
Order of Operations is a standard that defines the order in which you should simplify different
operations such as addition, subtraction, multiplication and division.
• Without it, two different people may interpret an equation or expression in different ways and
come up with different answers.
•The Order of Operations is shown below.
1) Parenthesis and Brackets -- Simplify the inside of parenthesis and brackets before you deal
with the exponent (if any) of the parenthesis or remove the parenthesis.
2) Exponents -- Simplify the exponent of a number or of a parenthesis before you multiply,
divide, add, or subtract it.
3) Multiplication and Division -- Simplify multiplication and division in the order they appear
from left to right.
4) Addition and Subtraction -- Simplify addition and subtraction in the order they appear from
left to right.
Example:
Basics of Word Problems
• A word problem in algebra is the equivalent of a story problem in math. When you
solved story problems in your math class you had to decide what information you had
and what you needed to find out. Then you decided what operation to use, if you needed
to find a total you would add, if you needed to find the change in something you
subtracted.
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•
•
•
In algebra you solve problems by essentially making two groups, one for each side of an
equation.
E.g. Sara has 15 apples and 12 oranges. How many pieces of fruit does she have?
We know that the sum of 15 and 12 is equal to the total amount of fruit. As you know or
read in our Basics of Algebra lesson, an unknown number or value is represented by a
letter. We don't know how many total pieces of fruit there are, so we can represent that
amount with x. On your paper you should define the variable by writing the following:
Direct and inverse relationship
• There are two kinds of relationships. If larger values of variable A cause larger values of
variable B, the relationship is direct. For example, as income rises, people spend more.
•
On the other hand, when larger values of variable A cause smaller values of variable B,
the relationship is inverse or indirect. For example, when the price of apples rise, people
will buy fewer
Example:
• 1) A and B are directly related. Initially, it is found that A=2 and B=49.18 If A is changed
to 2.8 what will be the corresponding value of B?
•
2) A and B are inversely related. Initially, it is found that A=2.9 and B=0.62 If B is
changed to 0.21 what will be the corresponding value of A?
•
•
3) Boyle's Law states that under conditions of constant temperature and quantity, there is
an inverse relationship between the volume and pressure for an ideal gas. A gas system
has an initial volume of 16. 0L with the pressure unknown. When the volume changes to
80.0L the pressure is found to be 715torr What was the initial pressure in torr?
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Lastly, percents can also be tied into ratios and proportions.
•Example: The ratio of salmon to walleye is 7 to 3. What is the percent of salmon?
Finite Math
• Graph of a Function The graph of a function f is the set of all points (x, f(x)) in the xyplane, where we restrict the values of x to lie in the domain of f.
• Example To get the graph of
– f(x) = 3x2 - 4x + 1 Function notation
• with domain restricted to [0, + ), we replace f(x) by y, getting the equation
– y = 3x2 - 4x + 1. Equation notation
• We then graph it by plotting points, restricting x to lie in [0, + ), and obtain the following
picture.
Linear Function
A linear function is a function of the form
f(x) = mx + b Function notation
y = mx + b Equation notation
with m and b being fixed numbers (the names 'm' and 'b' are traditional).
Role of m: If y = mx + b, then:
(a) y changes by m units for every 1-unit change in x.
(b) A change of x units in x results in a change of y = m x units in y.
(c) Solving for m, we obtain
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Straight Lines
The graph of a linear equation or function is a straight line.
The slope of the line through (x1, y1) and (x2, y2) is given by the formula
Sketching the Graph of a Linear Function
The easier way to sketch the graph of a linear function.
(a) Put the function in y = mx+b form, and draw the line with y-intercept b and slope m.
(b) Find the x- and y-intercepts and draw the line going through those two points. To find the xintercept of a line, set y = 0 in its equation and solve for x. To find the y-intercept, set x = 0 and
solve for y. This method works only if the line does not pass through the origin. If it does, then
you will need to plot an extra point or use the first method.
Examples
•Equation 2x - 3y = -6.
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Quadratic Equations