MATH SOLVE

4 months ago

Q:
# this 1 seems really complicated

Accepted Solution

A:

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:

We have:

→ " 4 (-4x + 16) − x = - 4 " ;

_________________________________________________

AND:

→ "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?

→ "0 = ? -4(4) + 16 " ?? ; → "0 = ? -16 + 16 ?? " ; → Yes! ;

{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!

_____________________________________________________

________________________________________________________

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:

We have:

→ " 4 (-4x + 16) − x = - 4 " ;

_________________________________________________

AND:

→ "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?

→ "0 = ? -4(4) + 16 " ?? ; → "0 = ? -16 + 16 ?? " ; → Yes! ;

{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!

_____________________________________________________