Find answers of this system of equations in reals {x+3y=4y^3, y+3z=4z^3, z+3x=4x^3

David Ali 2022-09-20 Answered
Find answers of this system of equations in reals
{ x + 3 y = 4 y 3 y + 3 z = 4 z 3 z + 3 x = 4 x 3
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Nathalie Rivers
Answered 2022-09-21 Author has 7 answers
Suppose that we had x > 1. Then, since 4 x 3 3 x > x, we have z > x, and similarly y > z, x > y, contradiction. By symmetry, we conclude that x , y , z [ 1 , 1 ].
So there exist α , β , γ [ 0 , π ] with x = cos α, y = cos β, z = cos γ. By the formula for cos 3 α, we can rewrite the system of equations as:
{ α ± 3 β   ( mod   2 π ) β = ± 3 γ   ( mod   2 π ) γ = ± 3 α   ( mod   2 π )
So we have ± 27 α α, so either 26 α 0 or 28 α 0. We find that α = π k / 13 or α = π k / 14 for some nonnegative integer k.
This gives 27 solutions, x = cos π k 13 for 0 k 13, and x = cos π k 14 for 1 k 13.
For example, one solution is ( ( cos π 14 , cos 9 π 14 , cos 3 π 14 )).
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2020-11-08
2x + 3y = 34
y = 5x
asked 2022-07-11
Sum of cubed roots
x 1 3 + x 2 3 + x 3 3
x 1 4 + x 2 4 + x 3 4
where x 1 , x 2 , x 3 are the roots of
x 3 + 2 x 2 + 3 x + 4 = 0
using Viete's formulas.
I know that x 1 2 + x 2 2 + x 3 2 = 2, as I already calculated that, but I can't seem to get the cube of the roots. I've tried
( x 1 2 + x 2 2 + x 3 2 ) ( x 1 + x 2 + x 3 )
but that did work.
asked 2022-05-23
For which value(s) of parameter m is there a solution for this system
{ m x + y = m m x + 2 y = 1 2 x + m y = m + 1
when does this system of equations have a solution?
asked 2022-10-27
The denominator of a fraction is 4 more than twice the numerator. When both the numerator and denominator are decreased by 6, the denominator becomes 12 times the numerator. Determine the fraction.
Let the numerator be 𝑥 and the denominator be 𝑦
Therefore, Fraction = x 4 + 2 x
Because, both the numerator and denominator are decreased by 6
Therefore, the new fraction becomes 𝑥 6 ( 4 + 2 𝑥 ) 6 = 12 𝑥
asked 2021-06-23

Write y=13x+7 in standard form using integers.

a. 2x+3y=21

b. 3x2y=21

c. 2x3y=21

asked 2022-10-08
Renata walks down an escalator that moves up and counts 150 steps. Her sister Fernanda climbs the same escalator and counts 75 steps. If the speed of Renata (in steps per time unit) is three times the speed of Fernanda, determine how many steps are visible on the escalator at any time.
asked 2022-11-18
Solving system of equations: x 3 3 y 2 x = 1 and 3 y x 2 y 3 = 1