Subtracting One Equation from Another
Date: 05/30/2001 at 23:18:03 From: Lee Subject: Subtracting one equation from another I'm 52. No one ever answered this question in school. I understand that if you have two equations with two variables - the same variables in each equation, say X and Y - there are two generally accepted ways of solving the equations. You can isolate one variable in one equation, then substitute it in the second equation. Or you can manipulate one of the equations so that you have equal values of one variable, then subtract one equation from the other, eliminating the variable. I can do the subtraction and solve the equations, but no one ever answered this question: Why can you subtract one equation from another? Not, why can you subtract one mixed variable from another, for example, 5X - 3X or 9X squared - 5X squared (I'm squaring the X). That's clear enough. But why can you subtract one equation from another and get something meaningful that you can use to solve the equations? The explanations I've heard are just given in more math jargon, which is no help. A side question. If you graph the equations, you get two lines. How can you subtract a line from another line? You can subtract the length of one line from the length of another, but how can you subtract lines? Lines are not numbers; they're marks on pieces of paper or chalkboards. There are points all along the lines, but no one says, "Subtract every point on the second line from every point on the first line." I still don't get it.
Date: 05/31/2001 at 09:28:05 From: Doctor Peterson Subject: Re: Subtracting one equation from another Hi, Lee. Great question! Let's do an example, just to have a specific example to discuss. We want to solve 2x + 5y = 26 x - 3y = 2 We double the second equation, giving 2x - 6y = 4 and then subtract the equations to get 0 + 11y = 22 so that y = 2. Why can we subtract one equation from another? If you picture an equation as a balanced scale, this is easy to explain: take the same amount off both sides of the scale, and it remains balanced. Here, "2x - 6y" is the same as 4; so we can subtract 4 from each side of the first equation (which will not change its meaning) by subtracting 2x-6y from the left side and 4 from the right. So the new equation will be true whenever both original equations are true. This gives us a simpler equation we can solve in order to find the solution to the original pair. Technically, we want to transform the problem into an equivalent one, that is, one with exactly the same set of equations; but the new equation by itself is not equivalent to both original equations: although any solution to the original pair will make this equation true, not all solutions to the new equation will be solutions of the pair. You can see this easily here, where x doesn't even appear in the new equation, so it can be anything. What's really happening is that the new equation _together with_ one of the pair is equivalent to the original pair. Now, what does this mean in terms of the graphs of the equations? Not much. Notice that if you multiply an equation by 2, as we might do before subtracting, we haven't changed its meaning at all - it's still the same line. So this is an action on the equation that has no effect on the graph. Since we can subtract any such form of the equation of one line from the other, you can see that the line itself doesn't determine what will happen when we subtract equations. I can't draw two lines and then say "this third line is the difference between those two lines," because there are many different possibilities for the result, depending on what equation we used for each line. In other words, you can't talk about subtracting two lines in this sense; you can only subtract equations. But you might like to try writing several equivalent equations for each of two lines, and then subtracting various pairs of equations and graphing them. You'll find that all the resulting lines have one thing in common: they all intersect at one point (assuming the original lines intersected). Two of these are used in solving the system: one that is horizontal, and gives us y, and one that is vertical, and gives us x. Now let's do one more thing: see how the two methods, subtraction and substitution, are equivalent. If we solved the same pair of equations by substitution, we might do this: 2x + 5y = 26 x - 3y = 2 From the second equation, x = 2 + 3y. Substituting in the first, 2(2 + 3y) + 5y = 26 Now, rather than solve this, I'm going to rearrange it to show that this is the same as subtracting twice the second equation from the first: 2(3y) + 5y = 26 - 2(2) 5y - 2(-3y) = 26 - 2(2) The x's have disappeared, which is our goal in either method; the left side is the 5y from the first equation minus twice the -3y from the second equation; and the right side is 26 minus twice the 2 from the second equation. So you could say that this method is just a different way to substitute. Let me know if you need any more help with these ideas. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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