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:
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} . ________________________________________________________
16 because 48/3 = 16 and we know that he started with 50 chocolates then distributed 3 times the number of friends present. This is written as 50 - 3x. We also know that there were 2 chocolates left so -3x = 2 - 50 which when simplified is x = 48/3 which equals 16.
