Identify the equivalent expression in the equation 1x2−x+1x=5x2−x and demonstrate multiplying by the common denominator.(1 point)
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
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
(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
(x−1)(1x^2−x)+(x−1)(1x)=(x−1)(5x^2−x)
To demonstrate multiplying by the common denominator, we need to find the common denominator of the fractions in the equation. The common denominator is (x^2−x).
Multiplying both sides of the equation by the common denominator:
(x^2−x)(x−1)(1x^2−x)+(x^2−x)(x−1)(1x)=(x^2−x)(x−1)(5x^2−x)
Simplifying each term:
(x−1)(x^2−x)+(x−1)(1x)=(x−1)(5x^2−x)
Expanding the parentheses:
x^3−x^2−x^2+x+x−1=x^3−x^2−5x^3+x^2+x
Combining like terms:
x^3−2x^2+x−1=−4x^3+2x
At this point, the equivalent expression in the equation has been identified and demonstrated by multiplying by the common denominator.
To identify the equivalent expression in the equation, we need to multiply both sides of the equation by the common denominator. In this case, the common denominator is (x^2 - x).
To demonstrate multiplying by the common denominator, we'll perform the multiplication on both sides of the equation step-by-step:
Starting with the equation:
1(x^2 - x) + 1(x) = 5(x^2 - x)
First, we'll multiply (x^2 - x) by 1/x to get rid of the denominator on the left-hand side:
(x^2 - x)(1/x) + (x^2 - x)(1) = 5(x^2 - x)
Next, we'll simplify the left-hand side:
(x^2/x - x/x) + (x^2 - x) = 5(x^2 - x)
This becomes:
(x - 1) + (x^2 - x) = 5(x^2 - x)
Finally, we'll multiply (x^2 - x) by 5 to get rid of the denominator on the right-hand side:
(x - 1) + (x^2 - x) = 5(x^2 - x)
Therefore, the equivalent expression in the equation is (x - 1)(1/x^2 - x) + (x - 1)(1/x) = (x - 1)(5x^2 - x).