Question

solve and check the equation.
\\(\frac{x}{8} = \frac{x}{9} - 9\\)

what is the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the solution set is \\(\square\\)
\\(\bigcirc\\) b. the solution set is \\(\\{x | x\text{ is a real number}\\}\\)
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\)

Explanation:

Step1: Eliminate denominators

Multiply all terms by the least common multiple of 8 and 9, which is 72:
$$72 \times \frac{x}{8} = 72 \times \frac{x}{9} - 72 \times 9$$
Simplify each term:
$$9x = 8x - 648$$

Step2: Isolate the variable term

Subtract $8x$ from both sides to collect $x$ terms on one side:
$$9x - 8x = 8x - 648 - 8x$$
Simplify:
$$x = -648$$

Step3: Verify the solution

Substitute $x=-648$ back into the original equation:
Left side: $\frac{-648}{8} = -81$
Right side: $\frac{-648}{9} - 9 = -72 - 9 = -81$
Both sides are equal, so the solution is valid.

Answer:

A. The solution set is $\{-648\}$