Question 46374
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(question 1)
What must be added to 5X+7Y-19Z to get 7X-7Y-15Z?

Let W be the answer.  Then 

5X+7Y-19Z PLUS W MUST EQUAL 7X-7Y-15Z
    <font face = "symbol">ß</font>      <font face = "symbol">ß</font>   <font face = "symbol">ß</font>     <font face = "symbol">ß</font>          <font face = "symbol">ß</font> 
5X+7Y-19Z  +   W     =      7X-7Y-15Z

   5X + 7Y - 19Z + W = 7X - 7Y - 15Z
 
Add -5X to both sides

   5X + 7Y - 19Z + W = 7X - 7Y - 15Z
  -5X                 -5X
 ------------------------------------
        7Y - 19Z + W = 2X - 7Y - 15Z

Add -7Y to both sides

        7Y - 19Z + W = 2X -  7Y - 15Z 
       -7Y                  -7Y
      --------------------------------
            -19Z + W = 2X - 14Y - 15Z 
         

            -19Z + W = 2X - 14Y - 15Z

Add +19Z to both sides

            -19Z + W = 2X - 14Y - 15Z
            +19Z                + 19Z
          -----------------------------
                   W = 2X - 14Y +  4Z

So the answer is 2X - 14Y + 4Z

=============================================

(question 2)
What must be subtracted from 3A+4B²-5C to give 2A+3B²+5C

Let W be the answer.  Then 

3A+4B²-5C MINUS W MUST EQUAL 2A+3B²+5C
    <font face = "symbol">ß</font>       <font face = "symbol">ß</font>   <font face = "symbol">ß</font>     <font face = "symbol">ß</font>         <font face = "symbol">ß</font> 
3A+4B²-5C   -   W     =      2A+3B²+5C

    3A + 4B² - 5C - W = 2A + 3B² + 5C
 
Add -3A to both sides

    3A + 4B² - 5C - W = 2A + 3B² + 5C
   -3A                 -3A
 ---------------------------------------
         4B² - 5C - W = -A + 3B² + 5C

Add -4B² to both sides

         4B² - 5C - W = -A + 3B² + 5C
        -4B²                -4B²
      ---------------------------------
              -5C - W = -A -  B² + 5C

Add +5C to both sides:

              -5C - W = -A - B² +  5C
              +5C                 +5C
            ----------------------------
                   -W = -A - B² + 10C       
 
We want W not -W, so we multiply both sides by -1

                -1(-W) = -1(-A - B² + 10C)         

                    W  = A + B² - 10C                   

So the answer is A + B² - 10

Edwin</pre>