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