find domain
x^2+9y^2/x-3y + 6xy/3y-x
To find the domain of the given expression, we need to determine the values that x and y can take which will not cause any division by zero or result in an undefined expression.
First, let's focus on the denominator of the first expression: x - 3y. For this expression to be defined, the value of x cannot be equal to 3y. This means that x ≠ 3y.
Next, let's look at the denominator of the second expression: 3y - x. Similarly, for this expression to be defined, the value of x cannot be equal to 3y. Therefore, x ≠ 3y.
Combining these two restrictions, we can conclude that x ≠ 3y for both expressions to be defined.
Other than that, there are no other restrictions on the variables x and y in the given expression. Therefore, the domain is all values of x and y except when x = 3y.
simplify
x^2+9y^2/x-3y + 6xy/3y-x
To simplify the expression
(x^2 + 9y^2)/(x - 3y) + (6xy)/(3y - x),
we can start by combining the fractions with a common denominator. The common denominator in this case is (x - 3y)(3y - x).
The first fraction, (x^2 + 9y^2)/(x - 3y), can be multiplied by (3y - x)/(3y - x), resulting in (x^2 + 9y^2)(3y - x)/[(x - 3y)(3y - x)].
The second fraction, (6xy)/(3y - x), can be multiplied by (x - 3y)/(x - 3y), which gives us (6xy)(x - 3y)/[(x - 3y)(3y - x)].
Combining the two fractions, we have:
[(x^2 + 9y^2)(3y - x) + (6xy)(x - 3y)]/[(x - 3y)(3y - x)].
Expanding the numerator, we get:
[3yx^2 - x^3 + 27y^3 - 9xy^2 + 6xy^2 - 18y^3]/[(x - 3y)(3y - x)].
Simplifying further, we have:
(3yx^2 - x^3 + 27y^3 - 3xy^2)/[(x - 3y)(3y - x)].
Therefore, the simplified expression is (3yx^2 - x^3 + 27y^3 - 3xy^2)/[(x - 3y)(3y - x)].
no
I apologize for the mistake. Let's start again.
The given expression is (x^2 + 9y^2)/(x - 3y) + (6xy)/(3y - x).
To simplify this expression, let's consider each fraction separately.
The first fraction can be factored as (x^2 + 9y^2) = (x + 3y)(x - 3y). Therefore, the first fraction becomes:
(x + 3y)(x - 3y) / (x - 3y).
The (x - 3y) in the numerator cancels out with the (x - 3y) in the denominator, leaving us with:
x + 3y.
Now, let's simplify the second fraction. We can factor out (-1) from the numerator to make it easier to combine with the first fraction:
6xy / (3y - x) = -6xy / (x - 3y).
The negative sign can be moved to the denominator:
-6xy / (x - 3y) = 6xy / (3y - x).
Now, we can combine the two simplified fractions:
x + 3y + 6xy / (3y - x).
Thus, the simplified expression is x + 3y + 6xy / (3y - x).
wrong
Apologies for the mistake. Let's try simplifying the expression again:
The given expression is (x^2 + 9y^2)/(x - 3y) + (6xy)/(3y - x).
To simplify this expression, we can try to find a common denominator and combine the fractions.
The common denominator is (x - 3y)(3y - x).
For the first fraction, we can multiply the numerator and denominator by (3y - x):
(x^2 + 9y^2)(3y - x)/[(x - 3y)(3y - x)].
Expanding the numerator, we get:
(3xy^2 - x^3 + 27y^3 - 9xy^2)/[(x - 3y)(3y - x)].
For the second fraction, we can also multiply the numerator and denominator by (x - 3y):
(6xy)(x - 3y)/[(x - 3y)(3y - x)].
Simplifying the numerator, we have:
6x^2y - 18xy^2.
Combining the fractions, we get:
(3xy^2 - x^3 + 27y^3 - 9xy^2 + 6x^2y - 18xy^2)/[(x - 3y)(3y - x)].
Now, we can simplify the numerator by combining like terms:
(-x^3 + 6x^2y - 25xy^2 + 27y^3)/[(x - 3y)(3y - x)].
Therefore, the simplified expression is (-x^3 + 6x^2y - 25xy^2 + 27y^3)/[(x - 3y)(3y - x)].