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Variables and Mathematical Models - Section 1.1
Algebra is an area of mathematics that deals with problem solving and finding unknown values based upon other
given information.
A variable is a letter or symbol that represents a quantity.
We often use the letters x, y, and z for variables, but many other letters will be used.
An algebraic expression is a combination of variables and numbers using the operations of addition, subtraction,
multiplication, division, powers, and roots.
Examples:
2x
y3
4
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Evaluating Algebraic Expressions
For now, we will start with the following agreed upon order of performing operations. We will add to this list
later.
Order of Operations:
1. Perform all operations inside grouping symbols (parenthesis, brackets, etc.)
2. Do all multiplication and division as they occur left to right.
3. Do all addition and subtraction as they occur left to right.
Examples: Evaluate each expression as indicated
1. 3 x
4 for x
2
1
2. 5x
3.
7y
2 and y
1
for x
6 and y
8
4
x
2x
2 for x
y
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Translating Algebraic Expressions
One of the main skills you will need to master in algebra is the ability to translate word phrases into algebraic
expressions. It is a good idea to know the key phrases that indicate the operations of addition, subtraction,
multiplication, and division. Here are some:
Operation
Key Words:
Addition
Subtraction
Multiplication
Division
plus
minus
times
divide
sum
difference
product
quotient
more than
less than
twice
ratio
increased by
decreased by
multiplied by
divided by
Examples: Write the following phrases into an algebraic expression. Let the variable x represent the number.
1. The sum of a number and 8
2
2. Five less than a number
3. the product of 7 and a number
4. Four more than twice a number
Pay close attention to the order of the words!!!
5. A number decreased by 9
6. A number subtracted from 9
7. Nine less than a number
8. Nine less a number
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Equations
An equation is composed of two algebraic expressions with an equal sign between them.
Solutions to an equation are values of the variable that make the equation a true statement (we often say the
solution satisfies the equation).
To determine if a number is a solution to an equation, simply replace the variable with that number and evaluate
both expressions on either side of the equation. If both sides evaluate to the same value, then the number is a
solution.
Examples: Determine whether the given number is a solution to the given equation.
1. 7x
2. 2 z
4
1
10; 2
3z
2; 1
Translate the following statements into algebraic equations. Let the variable x represent the number.
3. The quotient of a number divided by 6 is 5.
4. Six less than the product of 3 and a number yields 24.
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Watch for commas!
5. The product of 6 and a number increased by 5 is 30.
6. The product of 6 and a number, increased by 5 is 30.
Formulas and Models
A formula is an equation that expresses a relationship between two or more variables.
The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such
formula are often called mathematical models (or just models).
Based upon data from 1980 through 2010, the average price of a movie ticket, T, can be approximated by the
following formula
T
0. 15n
2. 72
where n is the number of years after 1980.
Find the average ticket price in 1990 using this model.
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