Question
Identify the equivalent expression in the equation 1x2−x+1x=5x2−x
and demonstrate multiplying by the common denominator.(1 point)
Responses
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
x2(1x2−x)+x2(1x)=x2(5x2−x)
x 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
(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−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
and demonstrate multiplying by the common denominator.(1 point)
Responses
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
x2(1x2−x)+x2(1x)=x2(5x2−x)
x 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
(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−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
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 left side is \(x(x^2 - x)\).
To eliminate the fractions, we multiply every term in the equation by the common denominator \(x(x^2 - x)\):
1. The left side has two terms:
- The first term \(\frac{1}{x^2 - x}\) becomes:
\[
x(x^2 - x) \cdot \frac{1}{x^2 - x} = x
\]
- The second term \(\frac{1}{x}\) becomes:
\[
x(x^2 - x) \cdot \frac{1}{x} = (x^2 - x)
\]
2. The right side \(\frac{5}{x^2 - x}\) becomes:
\[
x(x^2 - x) \cdot \frac{5}{x^2 - x} = 5x
\]
Thus, multiplying each term by the common denominator gives us:
\[
x + (x^2 - x) = 5x
\]
Now, analyzing the responses given:
1. **First response**:
\(x(1/x^2 - x) + x(1/x) = x(5/x^2 - x)\)
- This simplifies incorrectly.
2. **Second response**:
\(x^2(1/x^2 - x) + x^2(1/x) = x^2(5/x^2 - x)\)
- This simplifies incorrectly.
3. **Third response**:
\((x^2 - x)(1/x^2 - x) + (x^2 - x)(1/x) = (x^2 - x)(5/x^2 - x)\)
- This correctly employs the common denominator \(x^2 - x\) but is more complex than necessary as in the beginning we need \(x(x^2 - x)\).
4. **Fourth response**:
\((x - 1)(1/x^2 - x) + (x - 1)(1/x) = (x - 1)(5/x^2 - x)\)
- This also simplifies incorrectly.
Given the analysis, none of the provided responses perfectly aligns with the correct instruction of multiplying through by the common denominator \(x(x^2 - x)\).
The most straightforward equivalent expression reached by multiplying through the common denominator would be:
\[
x + (x^2 - x) = 5x
\]
However, if we must choose the most correct based on simplifications made without any incorrect premises in the methodology, we suggest the third response indicates an approach towards common denominators, albeit more complex.
So, the underlying operations would match the intention of multiplying by the common denominator, potentially leading toward justification for the right answers, but requires the simplest form involving the identified common denominator.
\[
\frac{1}{x^2 - x} + \frac{1}{x} = \frac{5}{x^2 - x}
\]
and demonstrate multiplying by the common denominator, we first identify the common denominator. The common denominator for the left side is \(x(x^2 - x)\).
To eliminate the fractions, we multiply every term in the equation by the common denominator \(x(x^2 - x)\):
1. The left side has two terms:
- The first term \(\frac{1}{x^2 - x}\) becomes:
\[
x(x^2 - x) \cdot \frac{1}{x^2 - x} = x
\]
- The second term \(\frac{1}{x}\) becomes:
\[
x(x^2 - x) \cdot \frac{1}{x} = (x^2 - x)
\]
2. The right side \(\frac{5}{x^2 - x}\) becomes:
\[
x(x^2 - x) \cdot \frac{5}{x^2 - x} = 5x
\]
Thus, multiplying each term by the common denominator gives us:
\[
x + (x^2 - x) = 5x
\]
Now, analyzing the responses given:
1. **First response**:
\(x(1/x^2 - x) + x(1/x) = x(5/x^2 - x)\)
- This simplifies incorrectly.
2. **Second response**:
\(x^2(1/x^2 - x) + x^2(1/x) = x^2(5/x^2 - x)\)
- This simplifies incorrectly.
3. **Third response**:
\((x^2 - x)(1/x^2 - x) + (x^2 - x)(1/x) = (x^2 - x)(5/x^2 - x)\)
- This correctly employs the common denominator \(x^2 - x\) but is more complex than necessary as in the beginning we need \(x(x^2 - x)\).
4. **Fourth response**:
\((x - 1)(1/x^2 - x) + (x - 1)(1/x) = (x - 1)(5/x^2 - x)\)
- This also simplifies incorrectly.
Given the analysis, none of the provided responses perfectly aligns with the correct instruction of multiplying through by the common denominator \(x(x^2 - x)\).
The most straightforward equivalent expression reached by multiplying through the common denominator would be:
\[
x + (x^2 - x) = 5x
\]
However, if we must choose the most correct based on simplifications made without any incorrect premises in the methodology, we suggest the third response indicates an approach towards common denominators, albeit more complex.
So, the underlying operations would match the intention of multiplying by the common denominator, potentially leading toward justification for the right answers, but requires the simplest form involving the identified common denominator.