Which ordered pair is in the inverse of the relation given by x^2y + 5y = 9?
(A) (2, 1)
(B) (-2, 1)
(C) (-1, 2)
(D) (2, -1)
(E) (1, -2)
y^2 x + 5 x = 9
if x = 1, y^2 = 4 so y =+/- 2
(1,-2) works
since (1,-2) satisfies the relation,
(-2,1) satisfies the inverse.
Oh, math problems! The bane of my existence as a Clown Bot. But fear not, for I shall give it a shot!
To find the inverse of the relation, we need to swap the x and y variables and solve for y. So let's embark on this math-filled adventure together!
First, let's call the relation f(x, y) = x^2y + 5y - 9.
If we swap the x and y variables, we get f(y, x) = y^2x + 5x - 9. Since we're looking for the inverse, we need to solve this equation for x.
Now, let's test out the given answer choices:
(A) (2, 1):
Plugging in these values, we get 1 = 1 + 5 - 9. Hmm, that's not quite right.
(B) (-2, 1):
Substituting these values in, we get 1 = 1 + 5 + 9. Nope, no good.
(C) (-1, 2):
Using these values, we get 8 = -2 + 10 - 9. Nope, not the one.
(D) (2, -1):
With these values, we get 1 = 2 - 5 - 9. Clearly not.
(E) (1, -2):
Plugging in these values, we get -2 = -2 + 10 - 9. Bingo! We've hit humor-filled mathematical gold!
So, the answer is (E) (1, -2).
Remember folks, when in doubt, laugh it out!
To find the inverse of a relation, we need to switch the roles of x and y and solve for y. Let's go step by step:
1. Start with the given relation: x^2y + 5y = 9.
2. Switch the roles of x and y: x^2y + 5x = 9.
3. Solve for y:
We'll isolate y by subtracting 5x from both sides of the equation:
x^2y = 9 - 5x.
Next, divide both sides of the equation by x^2:
y = (9 - 5x) / x^2.
Now, we have the equation for the inverse relation: y = (9 - 5x) / x^2.
To check which ordered pair is in the inverse, substitute the x-coordinate into the inverse equation and see if it matches the y-coordinate.
Let's check each option one by one:
(A) (2, 1):
Plugging x = 2 into the inverse equation:
y = (9 - 5(2)) / (2^2).
y = 1.
Since y = 1 matches the given y-coordinate, this ordered pair is a solution to the inverse relation.
(B) (-2, 1):
Plugging x = -2 into the inverse equation:
y = (9 - 5(-2)) / ((-2)^2).
y = -1.
Since y = -1 does not match the given y-coordinate, this ordered pair is not a solution to the inverse relation.
(C) (-1, 2):
Plugging x = -1 into the inverse equation:
y = (9 - 5(-1)) / ((-1)^2).
y = -4.
Since y = -4 does not match the given y-coordinate, this ordered pair is not a solution to the inverse relation.
(D) (2, -1):
Plugging x = 2 into the inverse equation:
y = (9 - 5(2)) / (2^2).
y = 1.
Since y = 1 does not match the given y-coordinate, this ordered pair is not a solution to the inverse relation.
(E) (1, -2):
Plugging x = 1 into the inverse equation:
y = (9 - 5(1)) / (1^2).
y = 4.
Since y = 4 does not match the given y-coordinate, this ordered pair is not a solution to the inverse relation.
Therefore, the only ordered pair that is in the inverse of the given relation is (A) (2, 1).