Strengthen Your Systems of Equations Skills

Find the values for $k$ that make this system:
- Inconsistent (no solutions)
- With unique solution
- With infinite solutions
$\{\begin{array}{l}kx+y+z=1\\ x+ky+z=1\\ x+y+kz=1\end{array}$

How to solve this equation?
$x=9+{\displaystyle \frac{y}{2}}$
$y=9+x$
$9+x+y=72$Solve $x$ and $y$

How do we solve the system of equations?
$\{\begin{array}{l}\sqrt[4]{x}({\displaystyle \frac{1}{4}}+{\displaystyle \frac{2\sqrt{x}+\sqrt{y}}{x+y}})=2\\ \sqrt[4]{y}({\displaystyle \frac{1}{4}}-<!--\; -\; -->{\displaystyle \frac{2\sqrt{x}+\sqrt{y}}{x+y}})=1\end{array}$

How to solve this system of ODE's?
$\left[\begin{array}{c}{\stackrel{\dot{}<!--\; \dot{}\; -->}{x}}_1\\ {\stackrel{\dot{}<!--\; \dot{}\; -->}{x}}_2\end{array}\right]=\left[\begin{array}{cc}\cos <!--\; \; -->t& -<!--\; -\; -->\sin <!--\; \; -->t\\ \sin <!--\; \; -->t& \cos <!--\; \; -->t\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

Prove that the system of equations below has only the solution $(x,y,z)=(1,1,1)$.
$\{\begin{array}{l}x+y^2+z^3=3\\ y+z^2+x^3=3\\ z+x^2+y^3=3\end{array}$

$$\begin{gathered} x+2y+z=5(x+y)(y+z) \\ x+y+2z=7(y+z)(z+x) \\ 2x+y+z=6(z+x)(x+y). \end{gathered}$$
Find the value of ${24}^{3}xyz$.

stability in the periodic orbit and in the singular point
$$\begin{gathered} \stackrel{\dot{}<!--\; \dot{}\; -->}{x}=-<!--\; -\; -->y+\mathrm{\lambda <!--\; \lambda \; -->}x(36-<!--\; -\; -->9x^2-<!--\; -\; -->y^2) \\ \stackrel{\dot{}<!--\; \dot{}\; -->}{y}=9x+\mathrm{\lambda <!--\; \lambda \; -->}y(36-<!--\; -\; -->9x^2-<!--\; -\; -->y^2) \\ \stackrel{\dot{}<!--\; \dot{}\; -->}{z}=-<!--\; -\; -->6z-<!--\; -\; -->{\mathrm{\lambda <!--\; \lambda \; -->}}^2{x}^2{y}^2{z}^3 \end{gathered}$$
I want to analyze the stability in the periodic orbit and in the singular point, so for the singular point I take the derived matrix of the linear part, and I got the eigenvalues, wich are ${\mathrm{\lambda <!--\; \lambda \; -->}}_1=-<!--\; -\; -->6$, ${\mathrm{\lambda <!--\; \lambda \; -->}}_2=3i$ , $\mathrm{\lambda <!--\; \lambda \; -->}=-<!--\; -\; -->3i$ . I wanted to use Andronov-Vitt but I have two eigenvalues with no real part so I can´t, does anybody can help me with this?

Finding the amount of solutions in a 3 equation solution
$x+2y+3z=7$
$3x+4y-<!--\; -\; -->z=4$
$3x+2y-<!--\; -\; -->11z=-<!--\; -\; -->13$

Solve the following system:
$\{\begin{array}{l}a{x}_0^2=\exp <!--\; \; -->(-<!--\; -\; -->{\displaystyle \frac{x_0^2}{4{\mathrm{\sigma <!--\; \sigma \; -->}}^2}})+a{r}^2\\ \exp <!--\; \; -->(-<!--\; -\; -->{\displaystyle \frac{x_0^2}{4{\mathrm{\sigma <!--\; \sigma \; -->}}^2}})+4a{\mathrm{\sigma <!--\; \sigma \; -->}}^2=0\end{array}$
where the varaibles are $x_0,a$. Take $x_0>0$

Find numbers $a$, $b$, $p$ and $q$ so that:
$x^3+15x^2+3x+5=p(x-<!--\; -\; -->a{)}^3+q(x-<!--\; -\; -->b{)}^3$

Solving a linear system of equations
$\{\begin{array}{l}3x-<!--\; -\; -->2y+z=8\\ 4x-<!--\; -\; -->y+3z=-<!--\; -\; -->1\\ 5x+y+2z=-<!--\; -\; -->1\end{array}$
Form two equations with $y$ elimanted.

Two equations & three unknowns (in$\mathbb{Z}$)
this system-equation has answer $(x,y,z)$ in Integers Set or not?
$a_{1}x+b_{1}y+c_{1}z=d_1$

Find the extreme values of the function
$f(x,y)=2(x-<!--\; -\; -->y{)}^2-<!--\; -\; -->x^4-<!--\; -\; -->y^4$
My attempt
$p=\frac{\mathrm{\delta <!--\; \delta \; -->}f}{\mathrm{\delta <!--\; \delta \; -->}x}=4(x-<!--\; -\; -->x^3-<!--\; -\; -->y)$
and
$q=\frac{\mathrm{\delta <!--\; \delta \; -->}f}{\mathrm{\delta <!--\; \delta \; -->}y}=4(y-<!--\; -\; -->y^3-<!--\; -\; -->x)$
Equating $p$ and $q$ to $0$ I get
$x-<!--\; -\; -->x^3-<!--\; -\; -->y=0$
and
$y-<!--\; -\; -->y^3-<!--\; -\; -->x=0$
I have no idea on how to solve the above $2$ equations

I'm trying to determine whether any solutions exist to a system of $(n+1)$ polynomial equations in $n$ unknowns. For example:
$\begin{array}{rl}xy& =-<!--\; -\; -->2\\ x^2-<!--\; -\; -->1& =0\\ y^3-<!--\; -\; -->3y^2+2y& =0\end{array}$
This is an accurate simplification of my actual system in that the first equation contains mixed terms with all variables represented and the remaining n equations are univariate, and that the maximum degree of any term (say, $D$) is in the low single digits. The actual system is very large ($n$ is ${10}^3$ to ${10}^4$), but with similar $D$. It also typically has many terms in the first equation.
In this example, the system is in fact consistent ($x=-<!--\; -\; -->1;y=2$). Were the first equation altered, however, to $xy=-<!--\; -\; -->3$, the system would be inconsistent. My naive approach is to find the solutions to the last n equations (here $x\in <!--\; \in \; -->\{-<!--\; -\; -->1,1\},y\in <!--\; \in \; -->\{0,1,2\}$) and try all combinations thereof until one satisfies the first equation or I exhaust the combinations, but this requires roughly $D^n$ checks, which isn't feasible for large systems. It's also not necessary to find a solution, just prove whether one exists.

Given the system of differential equations $x^{\prime }=2x+y^3$ and $y^{\prime }=-<!--\; -\; -->y$ i found the flow
${\mathrm{\varphi <!--\; \varphi \; -->}}_t(x,y)=((x_0+1/5y_0^3)e^{2t}-<!--\; -\; -->1/5y_0^3{e}^{-<!--\; -\; -->3t},y_0{e}^{-<!--\; -\; -->t})$

Find a basis for the solution set of the given homogeneous linear system
$3x_1+x_2+x_3=0$
$6x_1+2x_2+2x_3=0$
$-<!--\; -\; -->9x_1-<!--\; -\; -->3x_2-<!--\; -\; -->3x_3=0$
I do what I know I need to do. First I get the solution set of the system by reducing like this:
$\left(\begin{array}{ccc}3& 1& 1\\ 6& 2& 2\\ -<!--\; -\; -->9& -<!--\; -\; -->3& -<!--\; -\; -->3\end{array}\right)⇝<!--\; ⇝\; -->\left(\begin{array}{ccc}3& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)⇝<!--\; ⇝\; -->\left(\begin{array}{ccc}1& 1/3& 1/3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$
So I know $\stackrel{\to <!--\; \to \; -->}{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1-<!--\; -\; -->\frac{1}{3}r-<!--\; -\; -->\frac{1}{3}s\\ r\\ s\end{array}\right]$
That being the general solution.
Now, giving the values for $r$ and $s$ according to the standard vectors $i$, $j$
$\stackrel{\to <!--\; \to \; -->}{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1-<!--\; -\; -->\frac{1}{3}r-<!--\; -\; -->\frac{1}{3}s\\ r\\ s\end{array}\right]=r\left[\begin{array}{c}\frac{2}{3}\\ 1\\ 0\end{array}\right]+s\left[\begin{array}{c}\frac{2}{3}\\ 0\\ 1\end{array}\right]$
From my results, the basis will be:
$(\left[\begin{array}{c}\frac{2}{3}\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}\frac{2}{3}\\ 0\\ 1\end{array}\right])$
But instead, the book answer
$(\left[\begin{array}{c}-<!--\; -\; -->1\\ 3\\ 0\end{array}\right],\left[\begin{array}{c}-<!--\; -\; -->1\\ 0\\ 3\end{array}\right])$
Any idea on what I'm doing wrong?

How many solutions are there to this equation?
$\begin{array}{rl}{x}^2-<!--\; -\; -->y^2& =z\\ y^2-<!--\; -\; -->z^2& =x\\ z^2-<!--\; -\; -->x^2& =y\end{array}$

Steps to solve this system of equations:
$\sqrt{x}+y=7$
$\sqrt{y}+x=11$
Just wondering easiest step to find values for $x$ and $y$ from the above equations?

need to find four variables, $a,b,c,d,$, and I have four equations:
1) $u_{1}a-<!--\; -\; -->v_{1}b+c=u_1^{\prime }$
2) $u_{1}b-<!--\; -\; -->v_{1}a+d=v_1^{\prime }$
3) $u_{2}a-<!--\; -\; -->v_{2}b+c=u_2^{\prime }$
4) $u_{2}b-<!--\; -\; -->v_{2}a+d=v_2^{\prime }$
Note that $u_1,v_1,u_1^{\prime },v_1^{\prime },u_2,v_2,u_2^{\prime },v_2^{\prime }$ are all known.
Any suggestions to how to solve for the four variables, $a,b,c,d$?

Find the real values $x,y,z$ such that
$\{\begin{array}{l}x+y^2+z^3=21(1)\\ y+z^2+x^3=71(2)\\ z+x^2+y^3=45(3)\end{array}$