Question
Identify the equivalent expression in the equation 1x2−x+1x=5x2−x and demonstrate multiplying by the common denominator.(1 point)%0D%0AResponses%0D%0A%0D%0A(x2−x)(1x2−x)+(x2−x)(1x)=(x2−x)(5x2−x)%0D%0Aleft 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%0D%0A%0D%0A(x−1)(1x2−x)+(x−1)(1x)=(x−1)(5x2−x)%0D%0Aleft 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%0D%0A%0D%0Ax2(1x2−x)+x2(1x)=x2(5x2−x)%0D%0Ax squared left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus x squared left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals x squared left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis%0D%0A%0D%0Ax(1x2−x)+x(1x)=x(5x2−x)%0D%0Ax 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
Answers
Answered by
GPT-4o mini
To solve the equation \( \frac{1}{x^2 - x} - \frac{1}{x} = \frac{5}{x^2 - x} \) and demonstrate multiplying by the common denominator, we first identify the common denominator.
The common denominator for the fractions involving \( x^2 - x \) and \( x \) is \( x(x^2 - x) \).
Next, we multiply every term in the equation by this common denominator:
\[
x(x^2 - x) \left( \frac{1}{x^2 - x} \right) - x(x^2 - x) \left( \frac{1}{x} \right) = x(x^2 - x) \left( \frac{5}{x^2 - x} \right)
\]
Now we simplify each term:
1. For the first term:
\[
x(x^2 - x) \cdot \frac{1}{x^2 - x} = x
\]
2. For the second term:
\[
-x(x^2 - x) \cdot \frac{1}{x} = -(x^2 - x) = -x^2 + x
\]
3. For the right-hand side:
\[
x(x^2 - x) \cdot \frac{5}{x^2 - x} = 5x
\]
Now, substituting these results back into the equation yields:
\[
x - (x^2 - x) = 5x
\]
This simplifies to:
\[
x - x^2 + x = 5x
\]
Combining like terms leads to:
\[
- x^2 + 2x = 5x
\]
Rearranging gives:
\[
- x^2 + 2x - 5x = 0
\]
or
\[
- x^2 - 3x = 0
\]
This entire process illustrates how we can clear the fractions by multiplying by the common denominator.
Now, let's identify which response matches this multiplication by the common denominator.
None of the provided response options exactly match the derivation after clearing the fractions and multiplying by the common denominator. However, if we focus on the most relevant step regarding multiplying \( x^2 - x \) with fractions, it matches close to the correct form:
\[
\left( x^2 - x \right) \frac{1}{x^2 - x} + \left( x^2 - x \right) \frac{1}{x} = \left( x^2 - x \right) \frac{5}{x^2 - x}
\]
This corresponds to one of the first responses provided. Thus, the answer would be:
\[
(x^2 - x)\left(\frac{1}{x^2 - x}\right) + (x^2 - x)\left(\frac{1}{x}\right) = (x^2 - x)\left(\frac{5}{x^2 - x}\right)
\]
The common denominator for the fractions involving \( x^2 - x \) and \( x \) is \( x(x^2 - x) \).
Next, we multiply every term in the equation by this common denominator:
\[
x(x^2 - x) \left( \frac{1}{x^2 - x} \right) - x(x^2 - x) \left( \frac{1}{x} \right) = x(x^2 - x) \left( \frac{5}{x^2 - x} \right)
\]
Now we simplify each term:
1. For the first term:
\[
x(x^2 - x) \cdot \frac{1}{x^2 - x} = x
\]
2. For the second term:
\[
-x(x^2 - x) \cdot \frac{1}{x} = -(x^2 - x) = -x^2 + x
\]
3. For the right-hand side:
\[
x(x^2 - x) \cdot \frac{5}{x^2 - x} = 5x
\]
Now, substituting these results back into the equation yields:
\[
x - (x^2 - x) = 5x
\]
This simplifies to:
\[
x - x^2 + x = 5x
\]
Combining like terms leads to:
\[
- x^2 + 2x = 5x
\]
Rearranging gives:
\[
- x^2 + 2x - 5x = 0
\]
or
\[
- x^2 - 3x = 0
\]
This entire process illustrates how we can clear the fractions by multiplying by the common denominator.
Now, let's identify which response matches this multiplication by the common denominator.
None of the provided response options exactly match the derivation after clearing the fractions and multiplying by the common denominator. However, if we focus on the most relevant step regarding multiplying \( x^2 - x \) with fractions, it matches close to the correct form:
\[
\left( x^2 - x \right) \frac{1}{x^2 - x} + \left( x^2 - x \right) \frac{1}{x} = \left( x^2 - x \right) \frac{5}{x^2 - x}
\]
This corresponds to one of the first responses provided. Thus, the answer would be:
\[
(x^2 - x)\left(\frac{1}{x^2 - x}\right) + (x^2 - x)\left(\frac{1}{x}\right) = (x^2 - x)\left(\frac{5}{x^2 - x}\right)
\]