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 , Question from Berteanu, a student: I need help with this proposition: "It exists x a real number that for every y real number 5*x-2*y*y=1 This is false. Let x be from R. And I need an y real number that 5*x-2*y*y!=1 Please,could you help me?

Hi Berteanu,

Let's look at a similar statement that is true

There exists a real number $x$ so that for every real number $y, y - y = x.$

How would I prove this? I need to find an $x$ that works for any $y.$ Since it is to work for any $y$ let's try some value of $y$ say $y = 3.$ When $y = 3$ the equation $y - y = x$ becomes $0 = x.$ The question now is, if $x = 0$ is the equation $y - y = x$ true for any $y?$ That is, is it true that $y - y = 0$ for any $y?$ Of course it is so the statement "There exists a real number $x$ so that for every real number $y, y - y = x.$" is true and the value of $x$ is $x = 0.$

Now what about your statement "It exists x a real number that for every y real number $5 \times x - 2 \times y \times y = 1.$" What could the value of $x$ be? Let $y$ some real number and solve $5 \times x - 2 \times y \times y = 1$ for $x.$ Does this value of $x$ work for any $y?$ Try a different value of $y.$ Is the equation still true for this $y$ and the $x$ you found above?

Penny

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