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At first I thought my friend had pulled a fast one by scratching out the y's in the last step. But on second thought I realised it was correct. He did remind me that x is not equal to zero; since x = y, y is also non-zero. So, dividing both sides by y was perfectly okay; nothing wrong with that.
All those additions and subtractions were also okay because they were being carried out on both the sides.
So, everything was alright till the step: (x + y) * (x - y) = y * (x - y).
Then, he went and divided both sides by (x - y). That was all wrong. He shouldn't have done that; we know that x = y which means (x - y) = 0. So, he actually went and divided by 0! Division by 0 is undefined.
He had no business dividing both sides by (x - y) because (x - y) = 0. That was the error. Am I right?
The equation (x + y) * (x - y) = y * (x - y) is of the form A * C = B * C with A = (x + y), B = (x - y) and C = y. Now if A * B = B * C, then in order for the equality to hold, either A = C or B = 0. Now (x + y) is certainly not equal to "y" because if (x + y) is equal to y, then x has to be zero; but x is not equal to zero. Therefore, (x - y) = 0 has to be true; in this case, since both sides are multiplied by zero we have 0 = 0 which is true. So, (x - y) has to be zero. It is indeed true because we started with the statement x = y which means (x - y) = 0.
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- Union Public Service Commission - www.upsc.gov.in
- IIT-Kharagpur - www.iitkgp.ac.in
- Indian Statistical Institute - www.isical.ac.in
- Indian Institute of Technology Madras - www.iitm.ac.in
- Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
- Indian Institute of Mass Commission - www.iimc.nic.in
- IIT Bombay - www.iitb.ac.in
- Indian School of Mines, Dhanbad - www.ismdhanbad.ac.in
- Birla Institute of Technology, Ranchi - www.bitmesra.ac.in
- Central Institute of Fisheries Nautical and Engineering Training - www.cifnet.nic.in
- Indian Institute of Information Technology, Allahabad (Deemed University) - www.iiita.ac.in
- Central Marine Fisheries Research Institute, Kochi - www.cmfri.com
- Tata Institute of Social Sciences, Mumbai - www.tiss.edu