Question

I've been stuck on this problem for about 30 minutes now. Any help is greatly appreciated!!

Solve the equation.

(2/(z-5))-((6/4z+9))+((3/z-5))=0

Answer 1

is that (i am going to remove some parentheses)

2/(z-5)-6/(4z+9)+3/(z-5)=0

Answer 2

\[\frac{ 2 }{ z-5 }-\frac{ 6 }{ 4z+9 }+\frac{ 3 }{ z-5 } = 0\]

Answer 3

so before we do anything, notice that the left side expression is not defined if z = 5, or z = -9/4 , because of division by zero

Answer 4

right, since dividing by zero gives you an undefined

Answer 5

there are two ways to solve this, the easy way is to multiply both sides by the common denominator (z-5)*(4z + 9) , that will cancel the denominators

Answer 6

I tried it that way, but I multiplied it by the one on the left first and then the one on the right and that's where I got stuck

Answer 7

Multiply both sides by [(z-5) * (4z+9 )]

[(z-5) * (4z+9 )] * [2/(z-5)-6/(4z+9)+3/(z-5) ] =0*[(z-5) * (4z+9 )]

Did you get

2*(4z+9) - 6 ( z-5) + 3(4z+9) = 0

Answer 8

if not, i can explain how i canceled

Answer 9

No, I don't see how you got that

Answer 10

ok we need to distribute each term

Answer 11

the right side is clearly zero, since zero times anything is zero , agreed?

Answer 12

agreed

Answer 13

[(z-5) * (4z+9 )] * [2/(z-5)-6/(4z+9)+3/(z-5) ] = 0

distribute

[(z-5)*(4z+9)] * 2/(z-5) + [(z-5)*(4z+9)] *(-6)/(4z+9) + [(z-5)*(4z+9)]*3/(z-5) = 0

Answer 14

the general form of this is
[u ] * ( a - b + c ) = u*a + u*(-b) + u*c

Answer 15

so when you distribute the denominator it'll cancel out and you'll be left with distributing the top number?

Answer 16

right

Answer 17

for example when you multiply
[(z-5)*(4z+9)] * 2/(z-5)

the (z-5)/(z-5) cancels,

Answer 18

so after canceling, you should get
2*(4z+9) - 6 ( z-5) + 3(4z+9) = 0

Answer 19

Okay, I got there

Answer 20

ok now distribute

Answer 21

2*(4z+9) - 6 ( z-5) + 3(4z+9) = 0
distribute:
2*4z + 2*9 - 6*z -6*(-5) + 3*4z+3*9 = 0

Answer 22

8z+18-6z+30+12z+27=0

Answer 23

looks good

Answer 24

now collect like terms

Answer 25

14z+75=0

Answer 26

now solve for z

Answer 27

z=-75/14

Answer 28

right

Answer 29

now test this value, plug it back into the original equation

Answer 30

actually you can skip this part, (i just checked and its correct)

Answer 31

but you could if you wanted to :)

Answer 32

Thanks so much!!!