The solution to this system set is: "x = 4" , "y = 0" ; or write as: [4, 0] . ________________________________________________________ Given: ________________________________________________________ y = - 4x + 16 ;
4y − x + 4 = 0 ; ________________________________________________________ "Solve the system using substitution" . ________________________________________________________ First, let us simplify the second equation given, to get rid of the "0" ;
→ 4y − x + 4 = 0 ;
Subtract "4" from each side of the equation ;
→ 4y − x + 4 − 4 = 0 − 4 ;
→ 4y − x = -4 ; ________________________________________________________ So, we can now rewrite the two (2) equations in the given system: ________________________________________________________
y = - 4x + 16 ; ===> Refer to this as "Equation 1" ;
4y − x = -4 ; ===> Refer to this as "Equation 2" ; ________________________________________________________ Solve for "x" and "y" ; using "substitution" : ________________________________________________________ We are given, as "Equation 1" ;
→ " y = - 4x + 16 " ; _______________________________________________________ → Plug in this value for [all of] the value[s] for "y" into {"Equation 2"} ;
to solve for "x" ; as follows: _______________________________________________________ Note: "Equation 2" :
→ " 4y − x = - 4 " ; _________________________________________________ Substitute the value for "y" {i.e., the value provided for "y"; in "Equation 1}" ; for into the this [rewritten version of] "Equation 2" ; → and "rewrite the equation" ;
→ as follows: _________________________________________________
→ " 4 (-4x + 16) − x = -4 " ; _________________________________________________ Note the "distributive property" of multiplication : _________________________________________________
a(b + c) = ab + ac ; AND:
a(b − c) = ab − ac . _________________________________________________ As such:
→ "4 (-4x + 16) " = (4* -4x) + (4 *16) = " -16x + 64 " ; _________________________________________________ Now, we can write the entire equation:
→ " -16x + 64 − x = - 4 " ;
Note: " - 16x − x = -16x − 1x = -17x " ;
→ " -17x + 64 = - 4 " ; Solve for "x" ;
Subtract "64" from EACH SIDE of the equation:
→ " -17x + 64 − 64 = - 4 − 64 " ;
to get:
→ " -17x = -68 " ;
Divide EACH side of the equation by "-17" ; to isolate "x" on one side of the equation; & to solve for "x" ;
→ -17x / -17 = -68/ -17 ;
to get:
→ x = 4 ; ______________________________________ Now, Plug this value for "x" ; into "{Equation 1"} ;
which is: " y = -4x + 16" ; to solve for "y". ______________________________________
→ y = -4(4) + 16 ;
= -16 + 16 ;
→ y = 0 . _________________________________________________________ The solution to this system set is: "x = 4" , "y = 0" ; or write as: [4, 0] . _________________________________________________________ Now, let us check our answers—as directed in this very question itself ; _________________________________________________________ → Given the TWO (2) originally given equations in the system of equation; as they were originally rewitten;
→ Let us check;
→ For EACH of these 2 (TWO) equations; do these two equations hold true {i.e. do EACH SIDE of these equations have equal values on each side} ; when we "plug in" our obtained values of "4" (for "x") ; and "0" for "y" ??? ;
→ Consider the first equation given in our problem, as originally written in the system of equations:
→ " y = - 4x + 16 " ;
→ Substitute: "4" for "x" and "0" for "y" ; When done, are both sides equal?
{Actually, that is how we obtained our value for "y" initially.}.
→ Now, let us check the other equation given—as originally written in this very question:
→ " 4y − x + 4 = ?? 0 ??? " ;
→ Let us "plug in" our obtained values into the equation;
{that is: "4" for the "x-value" ; & "0" for the "y-value" ;
→ to see if the "other side of the equation" {i.e., the "right-hand side"} holds true {i.e., in the case of this very equation—is equal to "0".}.
→ " 4(0) − 4 + 4 = ? 0 ?? " ;
→ " 0 − 4 + 4 = ? 0 ?? " ;
→ " - 4 + 4 = ? 0 ?? " ; Yes! _____________________________________________________ → As such, from "checking [our] answer (obtained values)" , we can be reasonably certain that our answer [obtained values] : _____________________________________________________ → "x = 4" and "y = 0" ; or; write as: [0, 4] ; are correct. _____________________________________________________ Hope this lenghty explanation is of help! Best wishes! _____________________________________________________