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November 20th, 2010, 08:23 PM  #1 
Newbie Joined: Sep 2008 Posts: 4 Thanks: 0  Solution of y = x and y = x
I was helping someone with math and embarrassed myself today. I asked that person to use basic algebra to compute the distance between the line y = x and the point (4,4). I told the person to:
This process went pretty well until Step#3 above, where I embarrassed myself showing the person how to solve the linear system y = x y = x My first instinct was to solve the system by substituting the first equation y = x into the second equation y = x giving: x = x but this is complete nonsense right of the bat! How can x = x? If you divide both sides by x, then you get 1 = 1 which is even more bizarre. I guess I could try this: x = x x+x=x+x 2x=0 2x/2=0/2 x=0 ,but I still can't get over how x can ever be x. Much to my embarrassment, this blows my mind. Can someone please explain to me how I can solve the system y=x y=x without using a plotting or matrix algebra technique? Confused, bogger57 
November 20th, 2010, 09:46 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Solution of y = x and y = x
If you have y = x and y = x, you did correctly to set x = x Add x to both sides: 2x = 0 Divide through by 2 x = 0 Think of it as +0 = 0 
November 20th, 2010, 10:09 PM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407  Re: Solution of y = x and y = x Hello, bogger57! Quote:
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November 27th, 2010, 08:06 PM  #4 
Newbie Joined: Sep 2008 Posts: 4 Thanks: 0  Re: Solution of y = x and y = x
MarkFL and soroban: I can see how +0 = 0 so both of your answers seem to make sense to me, but I am still a little confused. Soroban says “You must NOT divide by x . . . ever!” I'm not sure what this means, since I don't remember my math teachers telling me I can't divide by x. Can someone show me what rule says I can't divide both sides of the following equations by x? yx = x x*x = x x+x+x+x=x+x+x Thanks for the Help So Far! Bogger 57 
November 27th, 2010, 08:21 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Solution of y = x and y = x
Using your examples: (1) If we divide through by x: As you can see y = 1 is a solution by the multiplicative identity, i.e , however, we miss a solution, namely x = 0. A better way to solve is to subtract x from both sides: Factor. Now we get both solutions, x = 0, y = 1. (2) Just like with (1) if we divide by x, we lose the solution x = 0. (3) If we divide by x, we get no solution, however if we subtract 3x from each side, we get: In general, if we divide by f(x), we lose the solution f(x) = 0. I was actually taught that if you do divide by f(x), note f(x) = 0 as a solution, then proceed. But strictly speaking, soroban is right, you should find another way to solve, finding all solutions at the end. 

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