Why study algebra? Because this topic provides the mathematical tools for any problem
more complicated than just combining some given numbers together.
Algebra lets you solve word problems in a regular and systematic way. Algebra le ts you
use symbols to solve all possible instances of a certain equation, not just a single example
of the equation with certain numbers in it. For example, it will help you understand the
properties of the universal equation for a straight line.
Really, What is Algebra?
Practically speaking, algebra has two main differences from the arithmetic youâ€™ve
studied so far:
1. We use variables as placeholders for unknown quantities.
2. We work on whole equations rather than just numbers.
What weâ€™re teaching in algebra is â€œhow you can torture equationsâ€ to get the information
you want. We will happen to do some arithmetic along the way, but the main idea is to
learn how to change and transform equations. Sometimes the information you want is a
simple number for a variable, and sometimes you want to know
how one variable is related to another.
It is common to have problems in which something is unknown. For example:
3 Ã— 4 = _____ or 7 + _____ = 10
In algebra, we change from using blanks or question marks to using letters or variables.
(Of course, the blanks above may be correctly filled in using 12 and 3, respectively!)
Using 'x' to stand for the first blank number and 'y' for the second we could write:
3 Ã— 4 = x and 7 + y = 10
These are really the same problems as the fill- in-the-blank problems above. In this case,
we could say the solution is x = 12 in the first equation and y = 3 in the second.
The letters are variables that are really just unknown numbers as was the blank. We can
use variables in any way that we could use other numbers. We can add and subtract,
multiply and even use them in powers and square roots.
There isn't anything special about 'x' and 'y', either. Although they are the most common
letters used, we can choose any letters we want. We will often choose letters that remind
us of the missing number. For instance, we might use h for height, l for length, v for
volume and t for time.
Letters are not reserved, and you really can choose what ever you like. But there are some
commonly used ones. The letters a, b and c are often used to indicate a constant. They
indicate a number of unknown value which is not variable; it is an unchanging number.
We just donâ€™t know what number it is.
A Short Form for Multiplication
Wait a minute! There could be some confusion with the letter x. Does it mean the
unknown number â€˜x' or to multiply? To avoid confusion, we have some new ways to
We can write the two multiplicands next to each other. If multiplication is not obvious,
we can use a dot â€¢ as a separator.
In fact, we will rarely use the symbol â€œÃ—â€ anymore:
|Â· Use a dot â€¢ between two numbers:
|â€œfour times sevenâ€
|Â· Or just write a number by a variable:
|â€œeight times tâ€
|Â· Write a value next to parentheses:
|4(x + 1)
|â€œfour times x plus oneâ€
|â€œtwo times threeâ€
|Â· Write a value next to an operation:
|â€œ4 times the square root of 25â€
|Â· Write to operations next to each other:
|â€œ3 factorial times 4 factorialâ€
|Â· Write parentheses next to each other:
|(x + 1)(x - 1)
|â€œx plus 1 times x minus 1â€
Neatness counts! You must make the dot â€¢ clearly different than a decimal point. Be
sure the reader can tell when you mean 4â€¢7 instead of 4.7!
Expressions are numbers combined together using various arithmetic operations.
For example, the expression 4 â€¢ 9 / 6 combines the numbers 4, 9 and 6 by multiplying 4
by 9 and then dividing by 6 giving the result 6.
Sometimes an expression will involve a variable. You can simplify the expression by
substituting the value of the variable into the expression and compute the result:
Find 3x - 6 if you are given that x = 7
|Put the value of 7 in for x
Multiply by 3 to get 21
Subtract 6 to get 15
|3 â€¢ 7 - 6
21 - 6