Question

solve and check the equation.
\\(\frac{x}{9} = \frac{x}{10} - 10\\)

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 \\(\\{\\ \\}\\).
\\(\bigcirc\\) b. the solution set is \\(\\{x \mid x\text{ is a real number}\\}\\).
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Find common denominator

The common denominator of 9 and 10 is 90. Multiply each term by 90 to eliminate fractions:
$$90\times\frac{x}{9}=90\times\frac{x}{10}-90\times10$$

Step2: Simplify each term

Simplify each term: \(10x = 9x - 900\)

Step3: Solve for x

Subtract \(9x\) from both sides: \(10x - 9x = -900\)
Which gives \(x = -900\)

Step4: Check the solution

Substitute \(x = -900\) into the original equation:
Left side: \(\frac{-900}{9} = -100\)
Right side: \(\frac{-900}{10} - 10 = -90 - 10 = -100\)
Both sides are equal, so \(x = -900\) is a valid solution.

Answer:

A. The solution set is \(\{-900\}\)