Algebra Review

When you read an equation, there is a mathematical order of precedence that tells you the order that you should perform operations. You may get different answers if you do the math in a different order. The following expression can be used to help you remember the order that the math should be performed:

Please Excuse My Dear Aunt Sally

The first letter of each word represents a mathematical operation:

P Parentheses

E Exponents

M Multiplication

D Division

A Addition

S Subtraction

Anything in parentheses should be done first. Once this is done, move on to any exponents (note: square root is the same as an exponent of 1/2). Next, do any multiplication; then division; then addition; and finally, subtraction.

Examples:

2 + 3 x 4 = ?
There are no parentheses and no exponents.
So the first step is to multiply 3 x 4 and get 12. Now we have:
2 + 12 = ?
All that is left is the addition to get a result of 14.

NOTE: Some calculators 'know' the order of precedence for math and others do not. Learn how your calculator works.

(2 + 3) x 4 = ?
There are parentheses so the first step is 2 + 3 = 5. Now we have:
5 x 4 = ?
Multiplying gives an answer of 20.

In the following example, each step will be shown with the explanation left to the viewer. Note: As you become more proficient with math, you can skip some steps.

(2 + 3)² + 4 x 2 = ?
5² + 4 x 2 = ?
25 + 4 x 2 = ?
25 + 8 = ?
Answer: 33

Reading Equations:

Algebra often uses symbols to represent variables. For example, suppose that your bank charges you $20 for each check that you bounce. To represent the amount your account is to be charged for bounced checks, they may write: 20B where B is the number of checks that you bounce. To find the total charge you multiply the $20 for each bounced check by the number of checks that your bounced (or that the bank says you bounced).

Although X is the most common variable name used in algebra, you can name your variables anything that makes sense (and you are not limited to one letter).

Algebraic equations usually consist of a combination of variables, coefficients, and scalars. For example, suppose we have the following equation:

Y = 3X + 6

Y and X are variables, 3 is a coefficient, and 6 is a scalar.

Subscripts:

In more advanced math, variables often have subscripts. For example, we may have a variable x_j (read "x sub j") or x_ij (read "x sub ij"). Consider the following example:

I want to predict the selling price of a house. To do this, I collect some data on past sales. The data I collect includes selling price, size of lot, number of square feet in the house, number of bedrooms, number of baths, and age of the house. A fairly traditional way to name the variables would be to let Y represent the quantity to be predicted and define X variables to correspond to the quantities used to help make the prediction. In this case, we would define Y as the selling price of a house, X₁ as the size of the lot, X₂ as the number of square feet in the house, X₃ as the number of bedrooms, X₄ as the number of baths, and X₅ as the age of the house. Note that we have defined the variables that will determine the data to be recorded for each house in the sample. Once we collect data we may want to refer to information for a specific house. That's were we could use the notation x_ij where x_ij represents the observed value for house i on variable j. Suppose the third house was on a 1.5 acre lot, had 2400 square feet, 4 bedrooms, 3.5 baths, was 5 years old, and sold for $175,000. For this we have x₃₁ = 1.5, x₃₂ = 2400, x₃₃ = 4, x₃₄ = 3.5, x₃₅ = 5 and y₃ = 175,000. It is also fairly traditional to refer to observed values of variables with lower case letters and to refer to the generic variable title with capital letters.

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