Question

solve.
$\frac{6z}{z-7} - \frac{42}{z+7} = \frac{588}{z^2 - 49}$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is {}. (simplify your answer. use a comma to separate answers as needed.)
b. the solution is the empty set.

Explanation:

Step1: Factor denominator

Notice that $z^2 - 49 = (z-7)(z+7)$

Step2: Multiply by common denominator

Multiply all terms by $(z-7)(z+7)$ to eliminate fractions:
$6z(z+7) - 42(z-7) = 588$

Step3: Expand and simplify

$6z^2 + 42z - 42z + 294 = 588$
Simplify like terms: $6z^2 + 294 = 588$

Step4: Isolate variable term

Subtract 294 from both sides:
$6z^2 = 588 - 294 = 294$

Step5: Solve for $z^2$

Divide both sides by 6:
$z^2 = \frac{294}{6} = 49$

Step6: Solve for z

Take square roots: $z = \pm 7$

Step7: Check for extraneous solutions

Original equation has denominators $z-7$, $z+7$, so $z
eq 7$ and $z
eq -7$. Both solutions make denominators zero, so they are invalid.

Answer:

B. The solution is the empty set.