Simplify. State the restrictions on the variable.
(6y-30/(y-1)) * (5y-5/3y^2-15y)
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(6y-30/(y-1)) * (5y-5/3y^2-15y)
=(6(y-5))/(y-1) * (5(y-1))/(3y(y-5))
=2*(5/y)
=10/y
There were two divisions, one by (y-1) , so y ≠ 1
the other was (3y(y-5)) , so y ≠ 0, 5
final answer would be
= 10/y , y ≠ 0, 1, 5
To simplify the given expression and state the restrictions on the variable, let's break down the expression step by step:
The given expression is:
(6y - 30/(y - 1)) * (5y - 5/3y^2 - 15y)
Step 1: Simplify each term within parentheses:
The first term can be factored out into 6 and (y - 1), giving us:
6(y - 1)
The second term can be simplified by finding the least common denominator (LCD) of 3y^2 and 15y, which is 3y^2. Multiplying the numerator and denominator of the fraction by the LCD, we get:
(5y * 3y^2 - 5 * 3y^2)/(3y^2) - 15y
Simplifying further, we have:
(15y^3 - 15y^2 - 45y)/(3y^2 - 15y)
Step 2: Combine the terms:
Now, we can multiply the two simplified expressions together:
6(y - 1) * (15y^3 - 15y^2 - 45y)/(3y^2 - 15y)
Step 3: Cancel out common factors:
Looking for common factors in the numerator and denominator, we can see that 15 is a common factor between the numerator and denominator. We can divide the numerator and denominator by 15 to cancel them out, resulting in:
2(y - 1) * (y^3 - y^2 - 3y)/(y^2 - 5y)
At this point, we are left with:
2(y - 1) * (y^3 - y^2 - 3y)/(y^2 - 5y)
To state the restrictions on the variable, we need to consider any values of y that would make the denominator zero since division by zero is undefined. In this case, to find the restrictions, we set the denominator equal to zero:
y^2 - 5y = 0
Factoring out y, we get:
y(y - 5) = 0
Therefore, the restrictions on the variable y are:
y = 0 and y = 5