# 1. For all real numbers x, there is a real number y such that 2x+y=7 would this be true or false?

Discrete math logic question
I have the following two questions.
1. For all real numbers x, there is a real number y such that $2x+y=7$ would this be true or false?I think true because if you put
$2\left(7\right)+y=14$
$2\left(8\right)+y=14$ there will always be a specific y that will make it work is this logic correct.
1. There is a real numbers x that for all real number y, $2x+y=7$ will be true.
would this be false because if you say $x=6$
then you get
$2\left(6\right)+2=14$
only if $y=2$ would it work but it would not work for every y.
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Anaya Gregory
Step 1
Yes, you're on the right track. Both statements can be more easily analyzed by rearranging to get
$x=\frac{14-y}{2}$
Step 2
This should allow you to fill in the gaps in your justification on (1) (i.e. not just stating a few examples in which it works).
###### Not exactly what you’re looking for?
Tamara Bryan
Step 1
Yes, indeed, you are correct in your assessment of the truth or falsity of each statement.
In the first, we can see this as allowing y to depend on x. So for any given x, we can find some y, and in particular, we can simply choose $y=7-2x$ which will guarantee the equalition holds.
Step 2
In the second case, y cannot depend on any given x. For the statement to be true, we need to consider the existence of a particular y such that for every x, regardless of what x may be, the equality holds. Since x can vary, but y can not vary accordingly, the statement is clearly false.
These two statements help demonstrate just how crucial the order of quantifiers and quantified variables can be: in the first, we have a true statement, and in the second, a false statement, and the only difference between them is the placement of $\mathrm{\exists }y\cdots$