The Divide between Numbers and Variables
Date: 04/25/2011 at 19:56:00 From: Brandon Subject: Diving 4x by 4 in 4x+7=5 In the equation 4x + 7 = 5, why do you divide 4x by 4? Why can't you just subtract 4 from 4x, like this? 4x - 7 = 5 4x = 12 -4 -4 x = 8 I guess it's supposed to be this: 4x - 7 = 5 4x = 12 4x/4 = 12/4 x = 3 But why? I have no idea about math. I slacked in school, but now I want to become more educated, so I'm going to start college.
Date: 04/25/2011 at 23:08:22 From: Doctor Peterson Subject: Re: Diving 4x by 4 in 4x+7=5 Hi, Brandon. This is an excellent question. I suspect a lot of students think it but don't ask it, and so they fail to learn this! When you subtract 4 from both sides of the equation 4x = 12, you get the new equation 4x - 4 = 12 - 4. This equation is equivalent to the original; that's not the problem. It's what you do next. The left side of this new equation, 4x - 4, is NOT equal to x. If that isn't obvious (because they are not like terms and can't be combined), just pick a value of x and see if the two expressions are equal. When x = 1, 4x - 4 is 0, which is not equal to x. So the two expressions are not equivalent, and you can't replace one with the other. The reason we divide by 4 here rather than subtract is to accomplish our goal of isolating x. The expression 4x/4 IS equal to x, because the division undoes the multiplication: when you multiply by 4, and then divide by 4, you get back the number you started with. So the general rule in solving an equation is that you must identify what operation you want to undo at each step, and then carry out that "undoing" operation. In your first step, you wanted to undo a subtraction of 7, and you did that by adding 7 (to each side). Now you want to undo a multiplication, so you divide. Another useful rule is to simplify the expressions on each side of the equals sign methodically and honestly. Don't just write "x" because that's what you WANT to see; actually simplify the expression so you'll realize that 4x - 4 is NOT x, and that 4x/4 IS. That would prevent a lot of students' mistakes! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 04/26/2011 at 00:06:32 From: Brandon Subject: Thank you (Diving 4x by 4 in 4x+7=5) Excellent explanation! Thank you for that. You answered my question -- and more.
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