I Solved It. Now What?
Date: 10/16/2003 at 15:56:37 From: Rachel Subject: algebra When I see an equation like 12 = 5x + x + 6 I have no idea what this means and don't have any idea on how to figure it out. Here's what I would try: 12 = 5x + x + 6 12 = 6x + 6 -6 -6 <-- Subtract 6 from 12 and from 6 -- ------ 6 6 <-- Divide 6 and 6x by 6 x = 1 Now what?
Date: 10/16/2003 at 16:12:50 From: Doctor Ian Subject: Re: I guess algebra Hi Rachel, You seem to know more than you think, because your solution was right on the money. Nicely done. Now let's think about what this final equation is telling us. Every time you made a change to an equation, it was a change that preserved the _meaning_ of the equation, right? So if you end up with x = 1 that means that 12 = 5x + x + 6 is really just a fancy way of saying that x is equal to 1! We can verify that by substituting 1 for x: 12 = 5x + x + 6 = 5*1 + 1 + 6 = 5 + 1 + 6 = 12 That is, when x has a value of 1, you get a true statement. For any other value you choose, the statement will be false. (Try some and see what happens.) Here's what you're doing, but in terms of language instead of equations. Suppose I start with a really complicated sentence like This belongs to the third son of the woman married to the man who is married to my mother. Now, my mother is married to my father. So the man who is married to my mother is my father, and I can shorten it to This belongs to the third son of the woman married to my father. But the woman married to my father is just my mother! This belongs to the third son of my mother. But I'm the third son of my mother: This belongs to me. At each stage, I made the language a little simpler, but I didn't change the meaning at all. So when I get to the end, I know that each sentence means the same thing as the one before it, which means that the first sentence and the last sentence are just two ways of saying the same thing. And that's exactly what's going on in algebra, when you transform a more complicated equation into a simpler one. This, by the way, is one reason that it's good to get into the habit of writing a series of transformed equations, rather than trying to work 'in place', as you were doing. For example, instead of doing this, 12 = 5x + x + 6 12 = 6x + 6 -6 -6 I'd go ahead and write a new equation, which I would then simplify: 12 - 6 = 6x + 6 - 6 6 = 6x Next, I'd do the division using a new equation, and simplify again: 6/6 = (6x)/6 1 = x This takes a little longer, but it's much easier to go back over your work and see that each step makes sense, by comparing each equation to the one before it. This is especially important if you're showing your work to someone else. So what's the point of doing all this? What can we do with the information that x is equal to 1? Well, in your math classes, the original equations are just made up by the authors of your textbooks to give you practice; but in real life, they arise from situations where we know something _about_ some quantity, but we don't yet know the quantity itself. So it's kind of like detective work: We start with a description like The murderer was a left-handed professional golfer who was born in Sweden between 1958 and 1972, walks with a limp, and drives a green Volkswagen Jetta. What we _want_ to do is get from that statement to one that looks like The murderer was Lars Edberg. What we _do_ with that information depends on why we wanted to know it in the first place. In the case of solving a murder, it's usually pretty clear why we want to know who did it. In the case of an equation, perhaps we want to know how many seats we should put in the airplane we're about to build, or how much money we should spend on advertising next year, or how much germanium we want to put in our next batch of semiconductors. It could be anything. But the idea is the same: We start with a description of a quantity, and want to determine its identity. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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