• Support PF! Buy your school textbooks, materials and every day products Here!

Solve for system of equations

  • Thread starter cupcakes
  • Start date
  • #1
18
0

Homework Statement


Given:
x7 + y7 =1
x + y = 1

Find the integer value(s) of (x-y)2.

Homework Equations


The Attempt at a Solution


I thought of substituting for y and then finding the rational roots but then I realized x and y don't have to be rational numbers for (x-y)^2 to be an integer. I am 90% sure the only solution is 1 (when x=1,y=0 or x=0,y=1) but don't know how to prove it. Any hint? Thanks.
 

Answers and Replies

  • #2
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,224
947

Homework Statement


Given:
x7 + y7 =1
x + y = 1

Find the integer value(s) of (x-y)2.

Homework Equations


The Attempt at a Solution


I thought of substituting for y and then finding the rational roots but then I realized x and y don't have to be rational numbers for (x-y)^2 to be an integer. I am 90% sure the only solution is 1 (when x=1,y=0 or x=0,y=1) but don't know how to prove it. Any hint? Thanks.
What are the solutions to the system of equations:

x7 + y7 =1

x + y = 1

?
 
  • #3
18
0
I think I'm very close to solving it. I substituted y= (1-x) in the first equation and expanded. Then I factored that. However I don't know how to factor the last term.

x4-2x3+3x2-2x+1

WolframAplha say it can be factored in to (x2-x+1)2. I just need to figure out how to factor that in to this and then I'm done (I think). Any ideas? Thanks.
 
  • #4
Mentallic
Homework Helper
3,798
94
I think I'm very close to solving it. I substituted y= (1-x) in the first equation and expanded. Then I factored that. However I don't know how to factor the last term.

x4-2x3+3x2-2x+1

WolframAplha say it can be factored in to (x2-x+1)2. I just need to figure out how to factor that in to this and then I'm done (I think). Any ideas? Thanks.
How did you get to that expression?
Expanding (1-x)7 allows us to cancel the 1 and -x7 terms, so the highest power of x is 6, then after dividing through by x it should be a max power of 5.
 
  • #5
18
0
How did you get to that expression?
Expanding (1-x)7 allows us to cancel the 1 and -x7 terms, so the highest power of x is 6, then after dividing through by x it should be a max power of 5.
Exactly. After expanding and canceling out we have:

7x6 - 21x5 + 35x4 - 35x3 + 21x2 - 7x.

First I divided by (7x). Then I realized (x-1) is a factor (since x=1 is a zero). After long division I have:

(7x)(x-1)(x4 - 2x3 + 3x2 - 2x + 1)
 
  • #6
Mentallic
Homework Helper
3,798
94
Ahh ok so you already factored out the x=1 factor.

So it's a quartic and hence it must have 4 complex roots. But since all the coefficients are real, the complex roots must come in complex conjugate pairs, and when you expand out [itex](x-\alpha)(x-\beta)[/itex] where [itex]\alpha, \beta \inℂ[/itex] it must be equal to a quadratic with real coefficients.

So, with this we can deduce that the quartic must be able to be factorized into

[tex](x^2+ax\pm 1)(x^2+bx\pm 1)[/tex]

And expanding that, then equating coefficients we can deduce a=b=-1 and we need to take the positive of the [itex]/pm[/itex] operator.
 
  • #7
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,705
1,722

Homework Statement


Given:
x7 + y7 =1
x + y = 1

Find the integer value(s) of (x-y)2.

Homework Equations


The Attempt at a Solution


I thought of substituting for y and then finding the rational roots but then I realized x and y don't have to be rational numbers for (x-y)^2 to be an integer. I am 90% sure the only solution is 1 (when x=1,y=0 or x=0,y=1) but don't know how to prove it. Any hint? Thanks.
One way to proceed is to look for the intersection of the two graphs y = 1 - x and y = (1-x^7)^(1/7). For the latter: look at the graph y vs x for x^n + y^n = 1. When n = 1 you get the line x+y=1. When n = 2 you get the circle x^2 + y^2 = 1, a circle of radius 1 passing through the points (1,0) and (0,1). What happens if n > 2? Well, any point (x,y) in the interior of the first quadrant and on the circle x^2 + y^2 = 1 must lie to the left and below the curve y = f(x) for x^n + y^n = 1. This is because such a point on the circle has 0 < x < 1 and 0 < y < 1, so x^n < x^2 and y^n < y^2, hence x^n + y^n < 1. That means we need to increase x and/or y to bring the quantity x^n + y^n up to 1. In other words, for n > 2 the graph is outside the circle except at the ends (1,0) and (0,1). That means that the intersection of the graph with x + y = 1 is easy to ascertain.

RGV
 
  • #8
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,224
947
I think I'm very close to solving it. I substituted y= (1-x) in the first equation and expanded. Then I factored that. However I don't know how to factor the last term.

x4-2x3+3x2-2x+1

WolframAplha say it can be factored in to (x2-x+1)2. I just need to figure out how to factor that in to this and then I'm done (I think). Any ideas? Thanks.
Split the 3x2 into x2 + 2x2 .

x4-2x3+3x2-2x+1
=x4-2x3+x2 + 2x2-2x+1

=(x2-x)2 + 2(x2-x) + 1

...​
 

Related Threads for: Solve for system of equations

Replies
3
Views
1K
Replies
14
Views
844
Replies
4
Views
3K
  • Last Post
Replies
2
Views
953
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
3
Views
1K
Top