2
votes
3answers
416 views

How to solve a system of three nonlinear equation in a simple way

Given the system: $$ \begin{cases} x^2y^2+x^2z^2=axyz & \\ y^2z^2+y^2x^2=bxyz &\\ z^2x^2+z^2y^2=cxyz \end{cases} $$ The solution could be gotten in a very tedious way. Is ...
4
votes
2answers
41 views

Prove or disprove the system about $n$th power has only one solution $x=y=1$

$$\begin{cases}x^n+y^n=2\\x+y=2\end{cases}\;,\;n\in\mathbb{N}\;,\;x,y\in\mathbb{R}\;,\;n>2$$ I have tried to show that $\displaystyle y'=-\frac{x^{n-1}}{y^{n-1}}=-1$ $$......$$ therefore $x=y=1$ ...
2
votes
3answers
175 views

How to solve the following pair of non-linear equations

$x^3+y^3=152\\x^2y+xy^2=120$ Best regards !
2
votes
2answers
55 views

Have I done something wrong in solving the following pair of equations?

Question given: Solve,$3x^2-5y^2-7=0\\3xy-4y^2-2=0$ What I have done so far: $$ ...
2
votes
4answers
115 views

How to solve the following system of equations.

$2x^2-3xy+2y^2=2\frac{3}{4}\\x^2-4xy+y^2+\frac{1}{2}=0$ I tried all the methods that I know, but I could't isolate $x$ or $y$ to form one equation.
2
votes
2answers
87 views

Uniqueness of a solution of the system of equations

A friend asked me the following question several days ago, and we still do not have a solution. Prove that the system of equations below has only the solution $(x, y, z)=(1, 1, 1)$. $$ \begin{cases} ...
0
votes
2answers
88 views

Number of real solutions for the following set of equations? [closed]

How to solve the following set of equations for real values of $x,y$ and $z$? $$x^2-y^2=z$$ $$y^2-z^2=x$$ $$z^2-x^2=y$$ $(0,0,0)$ is an obvious one solution.
0
votes
3answers
126 views

Find all $x,y,z$ satisfying $xy=z-x-y$ and cyclic permutations

Find all ordered pairs $(x,y,z)$ real numbers, which satisfy the following system of equations: $$xy=z-x-y\\xz=y-x-z\\yz=x-y-z$$
0
votes
3answers
143 views

How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?

How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?
1
vote
0answers
73 views

How to solve this system equation of polynomials?

I have: $F(x) + G(x) = 1 + F(x)*M,$ $G(x) = T_{1}(x) + T_{2}(x) + ... + T_{N}(x)$ $F(x )x^{a_{i}} = T_{i}(x) \times C_{i}(x) + \sum_{j \leq N} T_{j}(x) \times P_{ji}(x)$ In which $M, N, a_{i}, ...
4
votes
2answers
2k views

Sum of cubed roots

I need to calculate the sums $$x_1^3 + x_2^3 + x_3^3$$ and $$x_1^4 + x_2^4 + x_3^4$$ where $x_1, x_2, x_3$ are the roots of $$x^3+2x^2+3x+4=0$$ using Viete's formulas. I know that ...
6
votes
3answers
692 views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...