Question

solve for z. if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions enter none.\\(\frac{z - 18}{z^2 - 2z - 8} = \frac{5}{z + 2} + \frac{1}{z - 4}\\)\\(z = \underline{}\\)

Explanation:

Step1: Factor the denominator

Factor \(z^{2}-2z - 8=(z + 2)(z - 4)\)
So the equation becomes \(\frac{z - 18}{(z + 2)(z - 4)}=\frac{5}{z + 2}+\frac{1}{z - 4}\)

Step2: Multiply both sides by \((z + 2)(z - 4)\) (note \(z

eq - 2,z
eq4\))
\(z-18 = 5(z - 4)+1\times(z + 2)\)

Step3: Expand the right side

\(z-18=5z-20 + z + 2\)

Step4: Simplify the right side

\(z-18 = 6z-18\)

Step5: Move all terms to one side

\(z-6z=-18 + 18\)

Step6: Combine like terms

\(-5z=0\)

Step7: Solve for \(z\)

\(z = 0\)
We check if \(z = 0\) makes the original denominators zero. The denominators are \(0^{2}-2\times0 - 8=-8
eq0\), \(0 + 2=2
eq0\), \(0-4=-4
eq0\), so \(z = 0\) is valid.

Answer:

\(0\)