Identify the equivalent expression in the equation 1x2−x+1x=5x2−x and demonstrate multiplying by the common denominator.(1 point)

Question

Responses

(x2−x)(1x2−x)+(x2−x)(1x)=(x2−x)(5x2−x)
left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis

x(1x2−x)+x(1x)=x(5x2−x)
x left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus x left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals x left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis

(x−1)(1x2−x)+(x−1)(1x)=(x−1)(5x2−x)
left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis

x2(1x2−x)+x2(1x)=x2(5x2−x)

Answer 1

To solve the equation \( \frac{1}{x^2 - x} - x + \frac{1}{x} = 5x^2 - x \), we want to identify the equivalent expression and demonstrate multiplying by the common denominator.

The common denominator for the fractions \( \frac{1}{x^2 - x} \) and \( \frac{1}{x} \) is \( x(x^2 - x) \) or simplified, \( x(x(x - 1))=x^2(x - 1) \).

Multiplying each term by the common denominator \( x(x^2 - x) \):

1. for \( \frac{1}{x^2 - x} \), multiplying gives: \[ x \cdot 1 = x \]
2. for \( -x \), multiplying gives: \[ -x \cdot x(x^2 - x) = -x^2(x^2 - x) = -x^4 + x^3 \]
3. for \( \frac{1}{x} \), multiplying gives: \[ (x^2 - x) \cdot 1 = x^2 - x \]
4. for \( 5x^2 - x \), multiplying gives: \[ (5x^2 - x) \cdot x(x^2 - x) = 5x^3(x - 1) - x^2(x^2 - x) = 5x^3 - 5x^2 - x^4 + x^3 \]

Putting all this together, we get:

\[ x + (-x^4 + x^3) + (x^2 - x) = 5x^3 - 5x^2 - x^4 + x^3 \]

The equivalent expression that matches this process of multiplying by the common denominator is:

\[ (x^2 - x) \left( \frac{1}{x^2 - x} \right) + (x^2 - x) \left( \frac{1}{x} \right) = (x^2 - x) \left( 5x^2 - x \right) \]

Therefore, among the options provided, the correct choice is:

Response: \((x^2 - x) \left( \frac{1}{x^2 - x} \right) + (x^2 - x) \left( \frac{1}{x} \right) = (x^2 - x) \left( 5x^2 - x \right)\)