David The Maths Tutor

Welcome to the Unknown:

So now we know how to create new true equations from other true equations involving known numbers. But as you know, algebra is all about "solving for x" or "finding x". "x" is the most common letter used in algebra to represent a number that is unknown. Now other letters are also used, and the following discussion works with other letters as well.
Now suppose you had a brother Joe and a sister Tammy who gave you some money. You needed $10 and your siblings came to the rescue. It is now time to pay them back. You remember that Joe gave you $7, but you cannot remember what Tammy gave you. What Tammy gave you is, at the moment, an unknown number. Let's call that "x". You know that what Tammy gave you plus what Joe gave you is $10. The equation (sentence) in algebra that says this same thing is:
x + 7 = 10
Now I know you're all screaming the answer as this one can be done in your head without formal algebra - but this blog is about algebra, and the techniques here will help you solve equations that cannot be done in your head.
Now this equation is a true one and we can "solve for x" using the principle stated in the last post: you can create a new true equation from a true equation by doing the same arithmetic on both sides of the equation. It doesn't matter if you have an unknown number in the equation. The principle still applies.
So the objective is to "solve for x". This means use algebra to create an equation that looks like:
x = known number
So now look at x + 7 = 10. What can we do to get x by itself on the left side of the equation? What about subtracting 7 on both side?
x + 7 - 7 = 10 - 7 and then doing the arithmetic gives x + 0 = 3. But 0 added to anything is the same anything so it can be removed from the equation without changing the value of the left side. So we have
x = 3. We have solved for x. You can now stop screaming.
So the point of this post is that you can manipulate equations, even those with unknown numbers, as long as you do the same thing to both sides of the equation.

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Algebra: The Beginnings
As mentioned in my previous post, our math abilities are a by-product of other skills we developed as humans and these are used when we do language. Mathematics can be thought of as a language in its own right, though limited in subject matter, it is much more powerful and concise in expressing mathematical thoughts than any language we use when speaking.
Maths has its own set of words (symbols), its own dialects (scientists use a different set of symbols and equations than do engineers), and its own syntax and grammar. Algebra is the mathematical version of syntax and grammar, which is why it is such and important subject. It underpins all that is mathematics.
You are already familiar with many of the symbols (words) used in maths:
+ means "plus" or "positive" depending on context
− means "subtract" or "negative" depending on context
× means "multiplied by" or "times"
÷ means "divided by"
= means "equals"
So you already can read math sentences (equations) like 2 + 3 = 5 and you already know that there are rules (syntax) to be followed when writing an equation. For example, 2 3 + 5 = uses the same symbols, but it doesn't make sense.
Other equations are 3 = 3, 4 = 4, or 2 = 1. Just as in language, maths sentences can be true or false. But let's start with an obviously true sentence: 3 = 3. Now if I add 1 to both sides of the "=" sign, I get:
3 + 1 = 3 + 1 or 4 = 4, another true equation. Now let's divide both sides by 2:
4 ÷2 = 4 ÷2 or 2 = 2, Another true equation. I can do this all day, but let's just do one more:
2 - 2 = 2 - 2 or 0 = 0. In each step, notice that I had to do the same thing on both sides of the =. If there is only one algebra rule you should remember, it's this one: you can create a new true equation from a true equation by doing the same arithmetic on both sides of the equation. There are some caveats (cautions) regarding this, but this is the most important rule in algebra.

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