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)
1. x(1/x^2-x)+x(1/x)=x(5/x^2-x)
2. x^2(1/x^2-x)+x^2(1/x)=x^2(5/x^2-x)
3. (x-1)(1/x^2-x)+(x-1)(1/x)=(x-1)(5/x^2-x)
4. (x^2-x)(1/x^2-x)+(x^2-x)(1/x)=(x^2-x)(5/x^2-x)
The equivalent expression in the equation (1)/(x^(2)-x)+(1)/(x)=(5)/(x^(2)-x) is:
4. (x^2-x)(1/x^2-x)+(x^2-x)(1/x)=(x^2-x)(5/x^2-x)
To demonstrate multiplying by the common denominator, we multiply each term in the equation by x^2-x, which is the common denominator.
To identify the equivalent expression in the equation (1)/(x^(2)-x)+(1)/(x)=(5)/(x^(2)-x) and demonstrate multiplying by the common denominator, you would use option 4.
The common denominator for this equation is (x^2-x). Now, let's go through the steps of multiplying by the common denominator:
Starting with the equation: (1)/(x^(2)-x)+(1)/(x)=(5)/(x^(2)-x)
Multiplying each term by the common denominator (x^2-x), we get:
(x^2-x)(1/x^2-x) + (x^2-x)(1/x) = (x^2-x)(5/x^2-x)
Expanding and simplifying the equation, we get:
1 + x = 5(x^2-x)/(x^2-x)
Simplifying further, we have:
1 + x = 5
Thus, the equivalent expression in the equation is (x^2-x)(1/x^2-x) + (x^2-x)(1/x) = (x^2-x)(5/x^2-x).
To identify the equivalent expression in the equation and demonstrate multiplying by the common denominator, we need to find the common denominator for all the fractions on both sides of the equation. In this case, the common denominator is (x^2 - x).
Now let's go through the given options and see which one correctly demonstrates multiplying the equation by the common denominator:
1. x(1/x^2 - x) + x(1/x) = x(5/x^2 - x)
Option 1 doesn't correctly multiply the equation by the common denominator because the second term, x(1/x), is missing the common denominator (x^2 - x).
2. x^2(1/x^2 - x) + x^2(1/x) = x^2(5/x^2 - x)
Option 2 doesn't correctly multiply the equation by the common denominator because both terms have an additional factor of x.
3. (x - 1)(1/x^2 - x) + (x - 1)(1/x) = (x - 1)(5/x^2 - x)
Option 3 correctly multiplies the equation by the common denominator, (x^2 - x), and the expression on both sides of the equation.
4. (x^2 - x)(1/x^2 - x) + (x^2 - x)(1/x) = (x^2 - x)(5/x^2 - x)
Option 4 doesn't correctly multiply the equation by the common denominator because it multiplies the common denominator, (x^2 - x), by itself, resulting in (x^2 - x)^2.
Therefore, the answer is option 3:
(x - 1)(1/x^2 - x) + (x - 1)(1/x) = (x - 1)(5/x^2 - x)