Question

20 points!!!

Answer 1

The answer is:  {4, 1} .
________________________________________________________
Explanation:
________________________________________
Given the equations:
________________________________________
      "12x = 54 − 6y ;
     "-17x = -62 − 6y ;
_________________________________________
Multiply the second equation (both sides) by "-1" ;
___________________________________________
  -1*{-17x = -62 − 6y} ;

to get: 

17x = 62 + 6y ;
___________________________________________
{Note: The reason we do this is that we notice the TWO "-6y" values; and by multiplying one of the entire equations by "-1" ; we can change said equation to an equation with a "(+6y)" value; and the "(+6y)" and the "(-6y)" values cancel out to "zero" ; providing an opportunity to isolate "x"; and to solve for "x".}.
___________________________________________
Now, rewrite the two equations:
___________________________________________

       12x = 54 − 6y ;
       17x = 62 +  6y ;
___________________________________________
↔  Rewrite; and then add the two together:

       6y + 62 = 17x 
      -6y + 54 = 12x 
____________________
       0  +  116 = 29x ;

↔ 29x = 116 ;

Divide EACH SIDE of the equation by "29" ; to isolate "x" on one side of the equation ; and to solve for "x" ;
________________________________________________
   29x / 29 = 116 / 29  ;
________________________________________________
to get:
________________________________________________
           x  = 4 .
______________________________________________
Now that we have the value for "x" , which is "4" ; let us plug in "4" for "x" for either of the original two equations, to solve for "y".  In fact, let us try substituing "4" for "x" ;  for BOTH of the two original equations; to see if the value is correct.
_____________________________________________
Our original two given equations are:
_____________________________________________
      "12x = 54 − 6y ;

     "-17x = -62 − 6y ;
_____________________________________________
 Let us start with the first equation:
 ________________________
      " 12x = 54 − 6y " ;
__________________________________
When "x = 4" ; what does "y" equal ?

Plug in "4" for "x" ; to solve for "y" ;

  12(4) = 54 − 6y ;

→ 48 = 54 − 6y ;

Subtract "54" from EACH SIDE of the equation;

→ 48 − 54 = 54 − 6y − 54 ;

to get:

→  -6 = -6y ; 

→ Divide EACH side of the equation by "-6" ; to isolate "y" on one side of the equation ; and to solve for "y" ; 

→ -6/-6 = -6y / -6 ;

to get:

1 = y ;  ↔ y = 1 ;  So, we have:  x = 4, y = 1 ;  or,  {4, 1}.
______________________________________________
Let us check to see, if the second (orginal equation) holds true when "x = 4" and "y = 1 " ;
______________________________________________
The second "original equation" given is:
______________________________________________
       " -17x = -62 − 6y " ;
______________________________________________
   →    -17(4) = ?  -62  − 6(1) ?

    →  -68 = ? -62 − 6 ?

    →  -68 = ? -68 ?  Yes!
________________________________________________________
The answer is:  {4, 1} .
________________________________________________________