Question

solve the following equation. then, state whether the equation is an identity, a conditional equation, or an inconsistent equation.
\\(\frac{3}{x - 3}=7+\frac{x}{x - 3}\\)

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

Explanation:

Step1: Eliminate denominators

Multiply both sides by $x-3$ (note $x
eq 3$):
$$3 = 7(x-3) + x$$

Step2: Expand right-hand side

$$3 = 7x - 21 + x$$

Step3: Combine like terms

$$3 = 8x - 21$$

Step4: Isolate the variable term

Add 21 to both sides:
$$3 + 21 = 8x$$
$$24 = 8x$$

Step5: Solve for x

Divide by 8:
$$x = \frac{24}{8} = 3$$

Step6: Check validity

$x=3$ makes original denominator $x-3=0$, which is undefined.

Answer:

C. The solution set is $\emptyset$.