The most important distinctive feature of algebra is that it involves systems for figuring out ‘unknowns’. The way an unknown quantity is described in algebra is to use a letter instead of a number, for the obvious reason that it stands out because it is so different. The most commonly used letters are x & y, in their lower case forms. Believe it or not, they have a whole different meaning when used in their capitalized forms, and that will be explained further on.
The connection between algebra and arithmetic is strong and direct. Most of the time, you can describe a simple math problem using algebra. For example: A shop had 103 bath towels in stock on Monday. Whenever there are 55, it is time for the inventory clerk to order more. How many bath towels must be sold to trigger a new order? The answer to this question is ‘the x’. You can write this out in math sentences as: 103-x = 55. In this case, to solve for x, you just subtract 103-55 = x. Obviously, I have done something to the equations.
There is an important basic rule of algebraic equations. This is that whatever is on either side of the equation abides by the powerful ‘=’ sign: because of this you can manipulate an equation involving both sides to help solve for the variable. In the above case, I did 2 separate things. I subtracted 55 from both sides of the equation and I added an x to both sides. When working through these problems in a class or as a beginner it is wise and often helpful to do these moves separately. For example: we started with 103 – x = 55.
Step 1: Add 1x to each side of the equation. 103 = 55 + x. Step 2: Subtract 55 from both sides of the equation: 103-55 = x. Now use arithmetic to solve for x. x = 48. The shop has to sell 48 towels in order to justify ordering another batch of 103 from the manufacturer.
In this case, there is only one variable and there is only one correct answer. When this is the case, algebra is handy but not confusing. The biggest reason why people do sometimes find algebra ‘confusing’ is because many equations have multiple solutions, and many have vast numbers of answers; so many that you might get a google. Now, obviously, that’s overwhelming. What we do in these cases is to take a deep breath and to accept another principle of algebra: solution sets.
