Math CentralQuandaries & Queries


Question from Reasat, a student:

How to understand algebra?

Hi Reasat.

This is a big question in a few short words. Could you be a bit more specific about what you have trouble with? Perhaps there is a question on your homework or a quiz that you want to send us to describe how to solve it.

Algebra is mostly about identifying what you know about a problem, what you want to find out, and how these things are related. Algebra "names" things that have values with letters (we call them "variables") and relationships are written as equations involving these variables.

It's helpful to look at a simple example:

Q. Yasmina had 18 kg of pears to make pies to sell in the market. She found that she needed 1.5 kg of pears for each pie. How many whole pies could she make?

A. First, I will identify what I know: I know that she has a total of 18 kg of pears. So the total is 18. I choose to use the letter "t" to represent the total amount of pears. I also know that each pie requires 1.5 kg of pears for filling. So I choose "f" for the amount of filling in one pie. It is 1.5. Next, I see that the question asks for the number of pies she can make. I will let this be represented by the variable "p". And the final step in understanding the problem is to determine the relationship between the quantities. I know that the total (t) pears must be the same as (equals) the number of pies (p) times the amount of filling (f) in each pie.

So I have the formula t = p × f and I know what each of these represents.

Now I can insert the numbers I know: t = 18 and f = 1.5. I put parentheses around each of these to be extra careful. So

t = p × f is the same as
(18) = p × (1.5)

The last step in solving an algebra problem is to get the remaining variable (p in our case) by itself on one side of the equation. This is called "isolating the unknown variable".

The way I do it here is to recognize that I need to get rid of the 1.5 on the right hand side. The way to get rid of it in this case is to divide both sides by 1.5. Whatever you do to one side of the equation you must to do the other side. The reason I divide by 1.5 is that is the way to "undo" the multiplication by 1.5 that is already there. I think you can see that if you multiply by 1.5 and then divide by 1.5, you get the original value. Well, in our case the original value is p, which is what I want to know. So I have
(18) / (1.5) = p × (1.5) / 1.5

I simplify the left side by dividing:

12 = p × (1.5) / 1.5

and simplify the right side by canceling out the 1.5 values:

12 = p.

So this is saying that the answer is 12 pies!

I hope this demonstration helps you understand the process better. There are five steps:
1) Identify what you know (sometimes we say "what you are given"). Give them letter names.
2) Identify what you want to know (this is the answer). Give it a letter name.
3) Determine how the quantities relate to each other . Write this as an equation using the letter names.
4) Substitute in the values for the letters you know.
5) Isolate the unknown letter so it is on one side of the equal sign and just a number is on the other side.

Stephen La Rocque.

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