If Ax=B, Solve for x

Algebra may seem like a foreign language to many, but at its core it is really just a bunch of everyday operations, dressed up in the formal, and often offputting language of mathematics. Fortunately, in physics we can often be less formal and rigourous with our mathematics, so we can generally get by with an understanding of how to do algebra, without all the extra mathematical baggage. In this article I present the first, basic steps to algebra.

The basic idea of algebra is to find the unknown quantities, the 'x'es and 'y'es that make the equations you have true. In this article, I'll type the names of variables like x and y in italics to make them stand out. Despite its reputation, algebra can be solved by anyone – take an almost everyday example:

"I spent \$4.50 on three gallons of milk. How much was milk per gallon?"

Now, I imagine that most of you can automatically tell me that milk must have been \$1.50 per gallon. Let’s work backwards through the logic of how we come to this answer, so we can get a feel for how we do algebra.

If milk is \$1.50 per gallon, then three gallons is 3 . \$1.50 equals \$4.50 right?

Now, lets pretend we didn’t know the answer immediately – how would we proceed?

Let’s say that milk costs \$priceofmilk (In your math class, your teacher might have said "Let’s say that milk costs \$x" but I hope that calling the unknown by its name will make this clearer for you).

So, the price of three gallons is just three times \$priceofmilk. Or, writing this as an equation, we have:

\$4.50 = 3 . \$priceofmilk.
Now, looking at this equation, we have a statement that something (in this case \$4.50) is equal to something else (3x\$priceofmilk). What we want, is a statement that tells us that \$priceofmilk is equal to something else – hopefully \$1.50!

The right hand side of our equation, looks a little like what we want, but it has that troublesome three in it – in order to get rid of it, we need to divide by three

3 . \$priceofmilk / 3 = (3/3) . \$priceofmilk = 1 . \$priceofmilk = \$priceofmilk
However, if we want our equals sign to still be true, when we divide the right hand side by three, we need to divide the left hand side by three as well.
So, \$4.50 / 3 = \$priceofmilk = \$1.50
as we expected.

This principle of algebra is too basic to have a name (as far as I know – if you are aware of a name for it, please let me know). It can be generalized to many questions: For example, if I spent \$amountofmoney to buy numberofgallons of milk, what is the price of milk per gallon?

Setting this up the same way:

\$amountofmoney = numberofgallons . \$priceofmilk
now, to get \$priceofmilk by itself on the right hand side, we need to divide both sides by numberofgallons
\$priceofmilk = \$amountofmoney / numberofbottles

In more "classical algebra" terms, we could state this principle as:

If A=B.x where A and B just stand for some numbers that you are (or could, in principle be) given, then x=A/B