Identify the equivalent expression in the equation (1)/(x^(2)-x)+(1)/(x)=(5)/(x^(2)-x) and demonstrate multiplying by the common denominator. (1 point)
The common denominator for the equation is (x^2 - x).
To find the equivalent expression, we can multiply each term in the equation by the common denominator.
First, we'll multiply the left side of the equation:
((1)/(x^(2)-x)) * (x^2 - x) + ((1)/(x)) * (x^2 - x)
This simplifies to:
1 + (x^2 - x)/(x)
Next, we'll multiply the right side of the equation:
((5)/(x^(2)-x)) * (x^2 - x)
This simplifies to:
5
Therefore, the equivalent expression is:
1 + (x^2 - x)/(x) = 5
To find the equivalent expression in the equation, we need to multiply both sides by the common denominator. In this case, the common denominator is (x^2 - x).
So, let's multiply both sides of the equation by (x^2 - x):
(x^2 - x) * (1/(x^2 - x)) + (x^2 - x) * (1/x) = (x^2 - x) * (5/(x^2 - x))
Canceling out the common factors on each side, we get:
1 + (x^2 - x)/x = 5
Simplifying further, we can rewrite (x^2 - x)/x as (x - 1).
1 + (x - 1) = 5
Now, let's simplify the left side of the equation:
1 + x - 1 = 5
The +1 and -1 cancel out, leaving us with:
x = 5
So, the equivalent expression is x = 5.
To identify the equivalent expression in the equation and demonstrate multiplying by the common denominator, we first need to find the common denominator of the fractions involved.
Looking at the equation (1)/(x^(2)-x)+(1)/(x)=(5)/(x^(2)-x), the denominators are (x^2 - x) and x.
To find the common denominator, we need to factorize (x^2 - x) and determine if x is a factor.
Factoring (x^2 - x):
x(x - 1)
Now, we can see that x is a factor. So, the common denominator is x(x - 1).
Next, we multiply each fraction in the equation by the common denominator x(x - 1) to clear the denominators.
(x(x - 1))(1)/(x^(2)-x) + (x(x - 1))(1)/(x) = (x(x - 1))(5)/(x^(2)-x)
Simplifying each term, we have:
[x(x - 1)]/(x^2-x) + [x(x - 1)]/(x) = [5(x(x - 1))]/(x^2-x)
Now, the fractions have the same denominator, and we can add them together.
Combining the fractions, we get:
[x(x - 1) + x(x - 1)]/(x^2-x) = [5(x(x - 1))]/(x^2-x)
Simplifying the numerators:
[x^2 - x + x^2 - x]/(x^2-x) = [5x(x - 1)]/(x^2-x)
Combining like terms:
[2x^2 - 2x]/(x^2-x) = [5x(x - 1)]/(x^2-x)
Finally, we can see that the equivalent expression in the equation is:
[2x^2 - 2x]/(x^2-x) = [5x(x - 1)]/(x^2-x)
This is the result after multiplying both sides of the original equation by the common denominator, x(x - 1).