Which of the following expressions are equivalent to x - (-x) + y ?
choose all answers that apply:

a. 0 + y

b. y.

c. none of the above

 The correct answer choices are:

     [A]: " 0 + y "  ;   And:

     [B]:  " y " .

Step-by-step explanation:

We have:

The given expression:

" x - (-x) + y " .

We are asked:  "Which of the following expressions are equivalent to:

               "x - (-x) + y " ; {i.e. the given expression.}. ? "

 We are given 3 (three answer choices):

Choice  [A]:  " 0 + y " ;

Choice  [B]:  " y " ;

Choice  [C]:  "None of the above" .

Now, let us take a look at our given expression:

 " x - (-x) + y " .

Can this be simplified?  Yes!

The " x - (-x) " can be reduced to : " x + x " ; which equals:

                                                        " 1x + 1x " ;  which equals "2x" ;

and the "given expression" can be simplified to:

   " 2x + y " .

Note that:   " x - (-x) "  =  " x + x " ;

  because:  a "negative" minus a "negative" equals a "positive" ;

  that is;  the "negative value" of a "negative value" is a "positive value".

So we have:

  " 2x + y "  ;  as our "simplified" given expression.

Now, given our 3 (three) answer choices;  we are asked to choose all the answer choices that apply;

    →   {i.e. ;  {" Which of the 3 [three] answer choices are equal to the given expression;  or, for that matter, our "simplified" given expression:

          →   " x - (-x) + y "  ;    or;  " 2x + y " }  ?

Let us start with the first answer choice given:

    Choice  [A]:  " 0 + y " .

is:  " 2x + y " ;  equal to:  "0 + y " ?

Subtract "y" from each side of the equation:

   2x + y - y = 0 + y - y  ;  

to get:

   2x  = 0 ;  

Divide each side of the equation by "2" ;

   2x / 2 = 0 / 2 ;

to get:

x = 0 .

So, does:  " 2x + y  = 0 + y " ? ;

 Yes!  When:  x = 0 ! ;

Let us check by plugging in "0" for "x" ;

 " 2x + y =?  0 + y " ;

   →  " 2(0) + y =? 0 + y ;

Note:  " 2×0 = 0" .

   →  " 0 + y =? 0 + y " ?  Yes!

Now, let us check with the "non-simplified" given expression:

 " x - (-x) + y " ;

      →   " x - (-x) + y  = ?  0 + y " ? ;

Subtract "y" from each side of the equation;

      →   " x - (-x) + y - y  = ?  0 + y - y " ;

      →    "x - (x) =? 0 " ? ;

      →    " x - x = 0 " ? Yes! ;  {" 1x - 1x = 0 ."}.

So; Answer choice:  [A]: " 0 + y " ;  is a correct answer.

Let us continue with the second answer choice given:

    Choice  [B]:   " y " .

Is the simplified version of our given equation:  " 2x + y "  ;  

                   equal to:  "y " ?

Does:  " 2x + y =? y " ?

The answer is:  Yes;  because we have determined that:

 Answer choice:  [A]:  " 0 + y " ; is correct;  and:  

 Answer choice:  [B]:  " y " ; is equal to:  " 0 + y " .

As such: Answer choice:  [B]:  " y " ;  is a correct answer.

We can 'rule out" Answer choice:  [C]: "none of the above" ; since both of the above given answer choices are correct.

The correct answer choices are:

     [A]: " 0 + y "  ;   And:

     [B]:  " y " .

Hope this helps!

       Wishing you the best in your academic pursuits

               — and within the "" community!

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