The Idea Behind Simultaneous Equations
Date: 10/14/2001 at 14:52:30 From: Laura-Lee Subject: Systems of equations I'm having a hard time learning the linear combinations method and the substitution method in algebra II. Is there some kind of trick to memorizing them?
Date: 10/15/2001 at 12:22:37 From: Doctor Ian Subject: Re: Systems of equations Hi Laura-Lee, Actually, trying to memorize them would be about the worst thing you could do. The problem with memorizing things is that if you forget something that you've memorized, you leave yourself with no options except guessing what it's supposed to be. You'd be much better off trying to _understand_ how the various solution methods work. Better still, you can try to get a solid understanding of just what is going on when you have two simultaneous linear equations. Once you have that, you can choose from any of a number of possible solution methods, or even make up your own on the fly. When you have an equation like 3x + 2y = 5 there are infinitely many pairs of values for x and y that can make the equation true. For example, 3(1) + 2(1) = 5, so (x = 1, y = 1) is a solution to the equation. Also, 3(1/3) + 2(2) = 5 so ( x = 1/3, y = 2) is also a solution. An equation like this is a little like the volume control on a stereo. As you move the control to different positions (change the value of x), you increase or decrease the volume of the music (increase or decrease the value of y). Sometimes you can make the music as soft or as loud as you want. But there are other times when you have to meet some constraints. For example, perhaps you want to be able to hear the music in the next room (so it can't be too soft), but you don't want your mom to complain about it (so it can't be too loud). This is sort of like what happens when you have more than one equation, and all of them have to be true at the same time. For example, 3x + 2y = 5 and 5x + y = 13 As we've seen, if we consider either one of these equations by itself, it has an infinite number of possible solutions. But what if we consider them together? Then it turns out that there is only way to make _both_ equations true with a single pair of values, and that is to set x = 3 and y = -2. This can seem very mysterious, until you think about what's really happening here. Each of these equations describes a line. Each point on a line corresponds to a solution of the equation of the line. And if two lines aren't parallel, then they intersect at exactly one point. This point is the solution of the simultaneous equations. (If I tell you that family A is driving on Interstate X, and family B is driving on Interstate Y, they could be just about anywhere. But if I tell you that their cars collide, that tells you exactly where they were at the time of the collision!) When you think about it, this is a little like what you do with your eyes when you judge the distance to an object. An object that takes up a certain amount of space on your retina might be very small but very close; or very large but very far away; or somewhere in between. With one eye, you can't really tell which of these is the case, because there are infinitely many situations that could produce the same image on one retina: large small and and far? close? . . | . | | @ | | | One eye can't distinguish . | | among these possibilities... . | . or somewhere in between? That is, there is an 'equation' that relates a given size to a given distance, and it has infinitely many 'solutions'. But as soon as you bring a second eye into the picture, there will be only _one_ distance that 'solves' the 'equations' for both eyes at the same time: eye 2: "It's somewhere on this line" @ . . . . . x both eyes: "It must be here!" . . . . @ . eye 1: "It's somewhere on this line" That's the basic idea behind simultaneous equations. With that in mind, try reading about the combination and substitution methods again, and see if they make more sense to you. If they don't, write back and I'll try to find another way to help you understand why those methods work. I hope this helps. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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