Identify the equivalent expression in the equation, 1/x^2-x + 1/x = 5/x^2-x, and demonstrate multiplying by the common denominator.
The equivalent expression in the equation 1/x^2 - x + 1/x = 5/x^2 - x can be found by multiplying 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)
Expanding each term:
1 - x(x^2 - x) + (x^2 - x)(1/x) = 5(x^2 - x)
1 - x^3 + x^2 + x - x^2 + x = 5x^2 - 5x
Simplifying:
1 + x^2 + x = 5x^2 - 5x
The equivalent expression is 1 + x^2 + x = 5x^2 - 5x.
To find the equivalent expression in the equation 1/x^2 - x + 1/x = 5/x^2 - x, we need to multiply both sides of the equation by the common denominator.
The common denominator in this case is x(x^2 - x).
Multiplying both sides of the equation by x(x^2 - x), we get:
x(x^2 - x) * (1/x^2) + x(x^2 - x) * (1/x) = x(x^2 - x) * (5/x^2 - x)
Simplifying the equation step by step:
1 - x(x^2 - x)/x^2 + 1 - x(x^2 - x)/x = 5(x(x^2 - x))/x^2 - x(x^2 - x)
Now, let's simplify each term:
1 - x(x - 1)/x^2 + 1 - (x^2 - x) /x = 5(x^3 - x^2)/x^2 - x(x^2 - x)
Next, we combine like terms:
1 - (x^2 - x)/x^2 + 1 - (x^2 - x)/x = 5(x^3 - x^2)/x^2 - x(x^2 - x)
Now we can simplify further:
1 - (x^2 - x)/x^2 + 1 - (x^2 - x)/x = 5(x^3 - x^2)/x^2 - x^3 + x^2
Now we have the equivalent expression for the given equation by multiplying by the common denominator.
To identify the equivalent expression in the given equation 1/x^2 - x + 1/x = 5/x^2 - x, we need to create a common denominator.
In this case, the common denominator can be x(x^2 - x). To get this common denominator, we need to multiply the denominators of each fraction with the missing factor from the other fraction.
Let's do that step-by-step:
1) Multiply the first fraction, 1/x, by (x^2 - x)/ (x^2 - x):
(1/x) * (x^2 - x) = (x^2 - x) / (x^3 - x^2)
2) Multiply the second fraction, 1/x, by x/x:
(1/x) * x = x / x^2
After multiplying, we get:
(x^2 - x) / (x^3 - x^2) + x / x^2 = 5/x^2 - x
Now, we have a common denominator in all the fractions, which is x(x^2 - x):
[ (x^2 - x) + (x^3 - x^2) ] / (x^3 - x^2) = 5x / (x^3 - x^2)
Simplifying the numerator:
[x^2 - x + x^3 - x^2] / (x^3 - x^2) = 5x / (x^3 - x^2)
Combining like terms in the numerator:
(x^3 - x^2 - x^2 + x^2 + x) / (x^3 - x^2) = 5x / (x^3 - x^2)
The numerator simplifies to:
x^3 / (x^3 - x^2) = 5x / (x^3 - x^2)
Therefore, the equivalent expression in the equation is x^3 / (x^3 - x^2) and multiplying by the common denominator helped to simplify the equation.