For each of the following equations, determine which of the following statements are true:
(1) For all real numbers x, there exists a real number y such that the equation is true.
(2) There exists a realnumber x,such that for allreal numbers y, the equation is true.
Note that it is possible for both statements to be true or for both to be false.

A. 2x +y =7
B. y+x = x +y
C. x2 −2xy+y2 =0
D. (x −5)(y−1) =0
E. x2 +y2 =− 1

So, if you fix any value for x, you can choose y=7-2x and the equation will be true. The other claim is not true: the value for x that solves the equation depends on the value of y, so we can't hope to find a "universal" x-value such that the equation is true for all y-values.

B) Both are trivially true, because the equation is an identity: addition is commutative.

C) Only the first is true: we have

[tex]x^2-2xy+y^2=(x-y)^2[/tex]

So, whatever value we fix for x, we can choose y=x and the equation becomes

[tex](x-x)^2=0[/tex]

which is clearly true. The other claim is not true: the value for x that solves the equation depends on the value of y, so we can't hope to find a "universal" x-value such that the equation is true for all y-values.

D) Both claims are false: claim number 1 is false because the only solution to the equation is x=5, so you can't choose any value for x and expect to find a solution by choosing an appropriate value for y. Claim number 2 is false because, even if we choose x=5, the equation is true for all the possible values of y, except y=1, which is an excluded value for this expression

E) Both claims are false: [tex]x^2[/tex] and [tex]y^2[/tex] are positive numbers, because their are squares, and their sum can't be -1.