for what values of z does 4-z/4+z = z/z+z+z+z

Answer 1

(4-z)/(4+z) = (z/z) +z+z+z

3z + 1 = (4-z)/(z+4)

(3z+1)(z+4) = 4-z

3z^2 +13z+4 = 4-z

3z^2 +14z = 0

z(3z+14)= 0

3z = -14

z=-14/3

z =0, -14/3

but check the answers, z=0 make z/z undefined

so go with just x=-14/3

(14/3 +4)/(-14/3+4) = (-14/3)/(-14/3) - 14/3 -14/3-14/3

26/3/-2/3 = 1-14

-26/2 = -13

unless maybe you meant

(4-z)/(4+z) = z/(z+z+z+z)= z/4z

then

16z -4z^2= 4+z

4z^2 -15z +4 =0

(4z+1)(z-4) = 0

z= 4, -1/4,

but check the answers

for z=4

0/8 doesn't =4/16

for z=-1/4

(4+1/4)/(4-1/4) = -(1/4)/-1

17/4/15/4 = 1/4

17/15 doesn't = 1/4

solution is the Null Set, { }, DNE

extraneous solutions occur when you multiply by a variable

they are not solutions of the original equation

or maybe you meant

4+z/4-z = z/z + z+z+z

4 -3z/4 = 1 +3z

3= 15z/4

15z = 12

z = 12/15 = 4/5

check the answer

4+ (4/5)/4- 4/5 = (4/5)/(4/5) + 4/5+4/5+4/5

4+ 1/5 -4/5= 1+12/5

4-3/5 = 17/5

(20-3)/5=17/5