Question

type the correct answer in each box.
solve the equation, and enter the solutions from least to greatest. if there is only one solution, enter
.a.\ for the second solution.
\\(\frac{1}{x} + \frac{1}{x - 10} = \frac{x - 9}{x - 10}\\)
\\(x = \square\\) or \\(x = \square\\)

Explanation:

Step1: Eliminate denominators

Multiply each term by \(x(x - 10)\) (the least common denominator) to get rid of the fractions:
\(x(x - 10)\cdot\frac{1}{x}+x(x - 10)\cdot\frac{1}{x - 10}=x(x - 10)\cdot\frac{x - 9}{x - 10}\)
Simplify each term:
\((x - 10)+x=x(x - 9)\)

Step2: Simplify and solve

Combine like terms on the left:
\(2x - 10=x^{2}-9x\)
Rearrange into standard quadratic form:
\(x^{2}-11x + 10 = 0\)
Factor the quadratic:
\((x - 1)(x - 10)=0\)
Set each factor equal to zero:
\(x - 1 = 0\) or \(x - 10 = 0\)
But \(x = 10\) makes the original denominators zero (undefined), so we discard \(x = 10\).
Thus, \(x = 1\) and the second solution is n.a. (since \(x = 10\) is extraneous).

Answer:

\(x = 1\) or \(x = \text{n.a.}\)