The answer is: {4, 1} .
________________________________________________________
Explanation:
________________________________________
Given the equations:
________________________________________
"12x = 54 −<span> 6y ;
"-17x = -62 </span>− 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 <span>equation ; and</span> 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 <span>equation ; and</span> 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} .
________________________________________________________
________________________________________________________
Explanation:
________________________________________
Given the equations:
________________________________________
"12x = 54 −<span> 6y ;
"-17x = -62 </span>− 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 <span>equation ; and</span> 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 <span>equation ; and</span> 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} .
________________________________________________________
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