please help solve
- Thread starter ctvsurgn
- Start date
ctvsurgn said:
Are "Y" and "y" the same? Normally, they would be considered to be different variables!!
I think your problem is this:
x y = 42
x[sup:3icgu7p9]2[/sup:3icgu7p9] - y[sup:3icgu7p9]2[/sup:3icgu7p9] = 13
What is the value of (x - y)[sup:3icgu7p9]2[/sup:3icgu7p9]?
You can solve the first equation for y. If xy = 42, then y = 42/x
Substitute 42/x for "y" in the second equation:
x[sup:3icgu7p9]2[/sup:3icgu7p9] - (42/x)[sup:3icgu7p9]2[/sup:3icgu7p9] = 13
x[sup:3icgu7p9]2[/sup:3icgu7p9] - 42[sup:3icgu7p9]2[/sup:3icgu7p9] / x[sup:3icgu7p9]2[/sup:3icgu7p9] = 13
Multiply both sides of the equation by x[sup:3icgu7p9]2[/sup:3icgu7p9]
x[sup:3icgu7p9]4[/sup:3icgu7p9] - 42[sup:3icgu7p9]2[/sup:3icgu7p9] = 13x[sup:3icgu7p9]2[/sup:3icgu7p9]
x[sup:3icgu7p9]4[/sup:3icgu7p9] - 13x[sup:3icgu7p9]2[/sup:3icgu7p9] - 42[sup:3icgu7p9]2[/sup:3icgu7p9] = 0
Now, if you realize that 42[sup:3icgu7p9]2[/sup:3icgu7p9] = 6*7*6*7 it isn't too hard to factor this expression...
(x[sup:3icgu7p9]2[/sup:3icgu7p9] + 36)(x[sup:3icgu7p9]2[/sup:3icgu7p9] - 49) = 0
So, either x[sup:3icgu7p9]2[/sup:3icgu7p9] + 36 = 0, which means that x[sup:3icgu7p9]2[/sup:3icgu7p9] = -36 and x = +/- sqrt(-36) or x = +/- 6i,
OR
x[sup:3icgu7p9]2[/sup:3icgu7p9] - 49 = 0, or x[sup:3icgu7p9]2[/sup:3icgu7p9] = 49, or x = +/- 7
You now have four values of x...and that will also give you four values of y.
y = 42/x
so, y = 42/6i or 7/i
or y = 42/(-6i) or -7/i
or y = 42/7 or 6
or y = 42/(-7) or -6
With this assortment of values for x and y, it is just a matter of finding the value of
x[sup:3icgu7p9]2[/sup:3icgu7p9] - y[sup:3icgu7p9]2[/sup:3icgu7p9]
by substituting the various possible combinations.
I'm sure someone has a "neater" way of doing this problem, and I'm looking forward to seeing it!
mmm4444bot
Super Moderator
- Joined
- Oct 6, 2005
- Messages
- 10,962
Thank you for correcting your typographical error.
In order to find the value of (x - y)^2, we need to first find the values of numbers x and y.
We're told that the product x*y is 42. Since the product is positive, x and y must have the same sign.
We're also told that x^2 - y^2 = 13. The lefthand side of this equation is a difference of squares. Factor it.
Can you list the factors of 42 ?
There are only two factorization pairs (i.e., values of x and y) that work, in the following.
(x + y)(x - y) = 13
It takes less than a minute to test the possibilities, using brute force.
Re:
Aw gee...that makes it so easy!!
MrsPi
mmm4444bot said:
Thank you for correcting your typographical error.
In order to find the value of (x - y)^2, we need to first find the values of numbers x and y.
We're told that the product x*y is 42. Since the product is positive, x and y must have the same sign.
We're also told that x^2 - y^2 = 13. The lefthand side of this equation is a difference of squares. Factor it.
Can you list the factors of 42 ?
There are only two factorization pairs (i.e., values of x and y) that work, in the following.
(x + y)(x - y) = 13
It takes less than a minute to test the possibilities, using brute force.
Aw gee...that makes it so easy!!
MrsPi
mmm4444bot
Super Moderator
- Joined
- Oct 6, 2005
- Messages
- 10,962
Hi Mrs. Pi !
Our posts crossed in cyberspace. (Heh, heh)
Well, my way is easy, but I assumed Real x and y (and, thus, only one answer to the exercise).
(I never even considered non-Real values for x and y.
)You showed how there are two possible values for (x - y)^2, if we consider Complex values with an imaginary part for both x and y.
So, your approach is more complete.
Re:
What can I say? If there's a HARD way, that seems to be what I always do.
mmm4444bot said:
Hi Mrs. Pi !
Well, my way is easy, but I assumed Real x and y (and, thus, only one answer to the exercise).
(I never even considered non-Real values for x and y.
You showed how there are two possible values for (x - y)^2, if we consider Complex values with an imaginary part for both x and y.
So, your approach is more complete.
What can I say? If there's a HARD way, that seems to be what I always do.