Strengthen Your Systems of Equations Skills

3 lines 4 variables linear equation gaussian
$\{\begin{array}{l}\begin{array}{rl}& x+2y-<!--\; -\; -->z+3t=3\\ & 2x+4y+4z+3t=9\\ & 3x+6y-<!--\; -\; -->z+8t=10\end{array}\end{array}$

Is there a simpler approach to these system of equations?
I recently came across the following system of equations:
$$\begin{gathered} x+y+z=1 \\ {x}^2+y^2+z^2=2 \\ {x}^3+y^3+z^3=3 \end{gathered}$$
And I have two questions:
One, is there a way to prove or disprove whether there is a solution for this particular set of equations? Furthermore, is there a way to expand the proof for a more generalized set of equations, that is for this set:
$$\begin{gathered} x+y+z=1 \\ {x}^2+y^2+z^2=2 \\ ... \\ {x}^n+y^n+z^n=n \end{gathered}$$
Two, is there a simpler approach for the prior solution set than substitution?

Multiplication of two matrices refers to the composition of two linear transformations.
But, when we look at a matrix representation of systems of equations,
$Ax=b$
where $A$ is a m×n matrix, $x$ is $n\times <!--\; \times \; -->1$ matrix of $n$ variables, and $b$ is $m\times <!--\; \times \; -->1$ matrix. We apply matrix multiplication here, but we don't refer this to the composition of linear transformations since it is a representation of systems of equations.

Solving systems of equations from dynamics
Here is an example of a system I might have to solve for $F$, given $M$, $m$, ${\mathrm{\mu <!--\; \mu \; -->}}_S$, $\mathrm{\theta <!--\; \theta \; -->}$.
$N\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}-{\mathrm{\mu <!--\; \mu \; -->}}_{S}N\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=ma$
$N\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+{\mathrm{\mu <!--\; \mu \; -->}}_{S}N\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}-mg=0$
$-N\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+{\mathrm{\mu <!--\; \mu \; -->}}_{S}N\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+F=Ma$

How to solve this simultaneous equation of 3 variables.
$\begin{array}{}\text{(1)}& x+y+z& =& a+b+c\text{(2)}& ax+by+cz& =& a^2+b^2+c^2\text{(3)}& a{x}^2+b{y}^2+c{z}^2& =& a^3+b^3+c^3\end{array}$

What are the common solutions of $x^2+y=31$ and $y^2+x=41$?

systems of equations with 3 variables - addion method
$5x-<!--\; -\; -->y=3$
$3x+z=11$
$y-<!--\; -\; -->2z=-<!--\; -\; -->3$

$\begin{array}{}\text{(1)}& \frac{d^2\mathrm{\theta <!--\; \theta \; -->}}{dt}=\mathrm{\alpha <!--\; \alpha \; -->}(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1)+\mathrm{\beta <!--\; \beta \; -->}(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1{)}^3-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}\frac{d\mathrm{\theta <!--\; \theta \; -->}}{dt}\end{array}$
Where $\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->},\mathrm{\gamma <!--\; \gamma \; -->}\in <!--\; \in \; -->\mathbb{R}$.
This is my solution in attempting to convert (1) into a system of first order ODE's:
$\begin{array}{r}\text{(2)}& \frac{d\mathrm{\theta <!--\; \theta \; -->}}{dt}& =\mathrm{\varphi <!--\; \varphi \; -->}\\ \text{(3)}& \frac{d\mathrm{\varphi <!--\; \varphi \; -->}}{dt}& =\mathrm{\alpha <!--\; \alpha \; -->}(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1)+\mathrm{\beta <!--\; \beta \; -->}(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1{)}^3-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}\mathrm{\varphi <!--\; \varphi \; -->}\end{array}$
Is the above system correct? Also in equation (3) is it okay to have the $\mathrm{\theta <!--\; \theta \; -->}$ term?

Find fixed point
$\begin{array}{rl}{x}_{n+1}& :=0.5\cdot <!--\; \cdot \; -->(x_n^2+2\cdot <!--\; \cdot \; -->y_n-<!--\; -\; -->2\cdot <!--\; \cdot \; -->x_n\cdot <!--\; \cdot \; -->y_n-<!--\; -\; -->2\cdot <!--\; \cdot \; -->y_n^2)\\ y_{n+1}& :=-<!--\; -\; -->0.8\cdot <!--\; \cdot \; -->(x_n^2+2\cdot <!--\; \cdot \; -->y_n-<!--\; -\; -->2\cdot <!--\; \cdot \; -->x_n\cdot <!--\; \cdot \; -->y_n-<!--\; -\; -->2\cdot <!--\; \cdot \; -->y_n^2)\end{array}$
How to find $x$ and $y$?

Using the Lyapunov Function $V=\frac{1}{2}{x}_1^2+\frac{1}{2}{x}_2^2$, prove that the omega(ω)-limit set is non-empty for any initial value given for the dynamical system:
$x_1^{\prime }=x_1+2x_2-<!--\; -\; -->2x_1(x_1^2+x_2^2{)}^2$
$x_2^{\prime }=4x_1+3x_2-<!--\; -\; -->3x_2(x_1^2+x_2^2{)}^2$

Indetermined system
$\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}{x}_i=\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}\frac{1}{x_i}=3$
for $N>2$. The main aspect that confuses me is the general $N$ and I'm not sure how to proceed.

Solve for the variables w and x in the system of equations:
$\{\begin{array}{l}5w+4x=-<!--\; -\; -->2\\ 5w+2x=-<!--\; -\; -->3\end{array}$

Solve for the variables y and z in the system of equations:
$\{\begin{array}{l}6y+2z=-<!--\; -\; -->5\\ 4y+2z=3\end{array}$

Solve for the variables k and m in the system of equations:
$\{\begin{array}{l}6k+6m=-<!--\; -\; -->3\\ 2k+6m=-<!--\; -\; -->4\end{array}$

Solve for a and b: $\{\begin{array}{l}4a+6b=-<!--\; -\; -->2\\ 2a+4b=-<!--\; -\; -->1\end{array}$

Solve for j and k: $\{\begin{array}{l}2j+3k=-<!--\; -\; -->5\\ 6j+3k=2\end{array}$

Solve for the variables x and y in the system of equations:
$\{\begin{array}{l}4x+4y=3\\ 2x+6y=2\end{array}$

Solve the following linear system:
$x+2y-<!--\; -\; -->3z=4$
$3x-<!--\; -\; -->y+5z=2$
$4x+y+(s^2-<!--\; -\; -->14)z=s+2$

Arc length paramatrizations satisfy original system of differential equations?
Say we have a system of differential equations
$\{\begin{array}{l}{x}^{‴}(t)+f(t)x^{\prime }(t)=0\\ y^{‴}(t)+f(t)y^{\prime }(t)=0\end{array}$
on an interval $[a,b]$, along with the restriction that
$x^{\prime }(t{)}^2+y^{\prime }(t{)}^2=1$
on $[a,b]$.
Now, Euler proved that as long as $x(t),y(t)$ are continuous and not both zero at a point, then we can re-parametrize $x,y$ to some functions $\stackrel{~<!--\; ~\; -->}{x},\stackrel{~<!--\; ~\; -->}{y}$ which have the same image as $x,y$ but are arc-length parametrized.
Now, say we have a solution $(x^{\ast <!--\; \ast \; -->}(t),y^{\ast <!--\; \ast \; -->}(t))$ to
$\{\begin{array}{l}{x}^{‴}(t)+f(t)x^{\prime }(t)=0\\ y^{‴}(t)+f(t)y^{\prime }(t)=0\end{array}$
that satisfies the hypotheses needed to apply the previously mentioned re-parametrization. Say $({\stackrel{~<!--\; ~\; -->}{x}}^{\ast <!--\; \ast \; -->}(t),{\stackrel{~<!--\; ~\; -->}{y}}^{\ast <!--\; \ast \; -->}(t))$ are this re-parametrization, so they have the same image of $(x(t),y(t))$ on $[a,b]$ and satisfy
$x^{\prime }(t{)}^2+y^{\prime }(t{)}^2=1.$
Is it still the case that
$\{\begin{array}{l}({\stackrel{~<!--\; ~\; -->}{x}}^{\ast <!--\; \ast \; -->}{)}^{‴}(t)+f(t)({\stackrel{~<!--\; ~\; -->}{x}}^{\ast <!--\; \ast \; -->}{)}^{\prime }(t)=0\\ ({\stackrel{~<!--\; ~\; -->}{y}}^{\ast <!--\; \ast \; -->}{)}^{‴}(t)+f(t)({\stackrel{~<!--\; ~\; -->}{y}}^{\ast <!--\; \ast \; -->}{)}^{\prime }(t)=0?\end{array}$