Variables and Expressions

Variables are an important concept in algebra.
 
A VARIABLE is a placeholder, usually a letter, that we use to stand for a number or quantity. In algebra, we often use letters like "x" and "y" for variables, but we could use any of the 26 letters.
 
Most often a variable stands for an unknown number or quantity. But we can use variables to stand for any number - known or unknown.
 
Here's an example of a variable: We'll use "f" to stand for Tom's height in feet. If "f" is Tom's height in feet, what is his height in inches? The answer is 12 x f. There are 12 inches in a foot, and Tom's height in inches is 12 x f.
 
Since variables stand for numbers, we can add, subtract, multiply, and divide them just like we do numbers. We can also combine variables and numbers when we add, subtract, multiply, and divide. Let's say we wanted to add the variables x and y. It would look like this:

x + y

And if we added the variable 'y' and the number 4 it would look like this:

y + 4

When we multiply two numbers we use a sign like x or · to show that we are multiplying. For example, 5 times 7 could be written as 5x7 or 5·7. Or, we might use parentheses and write it like this: (5)(7).
 
When we want to multiply a number by a variable, for example 4 times the variable "r", we can simply write the product as: 4r. Similarly, we can write the product of the variables "x" and "y" as: xy.
 
Here are some other definitions:
 
An algebraic TERM is a number, a variable, or the product of a number and one or more variables.
 
An algebraic EXPRESSION is just the sum of a group of terms.
 
To EVALUATE an expression means to replace the variables with specific values. For example, if the value of x is 3, then the value of the expression "2x + 5" is 2(3) + 5 = 11.

Evaluate the following expression:

3x - 8x + 7ab

where x = 4, a = 6, b = -5

Let's evaluate the first term:

3x = 3 · 4 = 12

Next evaluate the second term:

-8x = -8 · 4 = -32

Then, evaluate the third term:

7ab = 7 · 6 · -5 = -210

Finally, add the terms:

12 - 32 - 210 = -230

So our final answer is -230.