Strengthen Your Systems of Equations Skills

How to solve $0=x\times <!--\; \times \; -->114-<!--\; -\; -->x\times <!--\; \times \; -->{\log }_3<!--\; \; -->(x)-<!--\; -\; -->20.28\times <!--\; \times \; -->y$ in matlab for different values of $y$?
$y={10}^3,{10}^6,{10}^9,{10}^{12},{10}^{15},...$ and above mentioned equation. How to solve (i.e. getting values of x for different y) and plot this equation in MATLAB ?

Solving system of equations: $x^3-<!--\; -\; -->3y^{2}x=-<!--\; -\; -->1$ and $3y{x}^2-<!--\; -\; -->y^3=1$

Solve following equations:
$\{\begin{array}{l}K=\frac{B–3}{20}\\ K=(20S+3)R+S\\ K=20S^2+(20N+7)S+N\\ N=S-<!--\; -\; -->R\end{array}$- And the $B$ values, e.g : $173,283,2343,834343$How to find the $R$ values?

An average mark is computed for 100 students in Business, an average is computed for 300 students in Arts, and an average is computed for 200 students in Science. The average of these three averages is 85%. However, the overall average for the 600 students is 86%. Also, the average for the 300 students in Business and Science is 4 marks higher than the average for the students in Arts. Determine the average for each group of students by solving a system of linear equations.
Let $x_1,x_2,x_3$ represent the students in business, arts and science respectively.

$100x_1+300x_2+200x_3=86$
$100x_1+200x_3+4=300x_2$Not sure as to how to create an equation for the first one.

$a:[0,\mathrm{\infty <!--\; \infty \; -->})\to <!--\; \to \; -->\mathbb{R}$ is a continous and bounded and
$x^{\prime }(t)=\left(\begin{array}{cc}0& 1\\ -<!--\; -\; -->a(t)& 0\end{array}\right)x(t)$
has a non-zero solution like $y(t)$ such that $\underset{t\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}y(t)=0$.
Show that this equation has an unbounded solution on $[0,\mathrm{\infty <!--\; \infty \; -->})$.

Find answers of this system of equations in real numbers
$\{\begin{array}{c}x+\frac{2}{y}=3\\ y+\frac{2}{z}=3\\ z+\frac{2}{x}=3\end{array}$

For what values of k and h does this system of equations have a unique solution?
$x-3y+2z=5$
$2x-5y-3z=9$
$-x-y+kz=h$
So $\left[\begin{array}{cccc}1& -<!--\; -\; -->3& 2& 5\\ \\ 2& -<!--\; -\; -->5& -<!--\; -\; -->3& 9\\ \\ -<!--\; -\; -->1& -<!--\; -\; -->1& k& h\end{array}\right]$
When row reduce: $\left[\begin{array}{cccc}1& 0& -<!--\; -\; -->19& 2\\ \\ 0& 1& -<!--\; -\; -->7& -<!--\; -\; -->1\\ \\ 0& 0& k-<!--\; -\; -->26& h+1\end{array}\right]$
1) has a unique solution.
2) has infinite number of solutions.
3) has no solution.

All the solutions for this system 5x+33y = 6 (mod 13) and 7x + 2y = 9 (mod 13)

Let's say we have the general solution to $X^{\prime }=A(t)X$, where $X=(x_1,x_2{)}^T$. How do you find the general solution to the system $X^{\prime }=A(t)X+b(t)$ where $b(t)$ is a $2\times <!--\; \times \; -->1$ matrix with two polynomials as entries. How do you find the particular solution?

Nonlinear system:
$\left(\begin{array}{c}{\stackrel{\dot{}<!--\; \dot{}\; -->}{y}}_1\\ {\stackrel{\dot{}<!--\; \dot{}\; -->}{y}}_2\end{array}\right)=\left(\begin{array}{c}2y_1\\ y_1^2\end{array}\right)$
Giving the jacobian of transformation being:
$\stackrel{\dot{}<!--\; \dot{}\; -->}{F}=\left(\begin{array}{cc}2& 0\\ 2y_1& 0\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 2& 0\end{array}\right)\left(\begin{array}{c}{F}_1\\ F_2\end{array}\right)+\left(\begin{array}{c}2\\ 0\end{array}\right)$
Which gives me the eigenvalues $\mathrm{\lambda <!--\; \lambda \; -->}=0^2=0,0$ meaning a degenerate node since we have only the eigenvector:
$F^{(1)}=\left(\begin{array}{c}0\\ 1\end{array}\right)$
Non-linear system with all trajectories converging on the line $x=0$, rather than $(2,0)$?

Generalized way of solving this types of equations $x^3+y^4=z^5$

What would be the process of solving a modular cubic equation?
$$a{x}^3+b{x}^2+cx+d=0(\mathrm{mod}n)$$

Solve equation with complex numbers using a helper equation
$z^2=5-<!--\; -\; -->12$
Let $\text{Let}z=x+yi$
$(x+yi{)}^2=5-<!--\; -\; -->12i$
$x^2-<!--\; -\; -->y^2+2ixy=5-<!--\; -\; -->12i$
This means that we get the following system of equations:
$\{\begin{array}{l}{x}^2-<!--\; -\; -->y^2=5\\ 2xy=12\end{array}$
$|z^2|=\sqrt{5^2-<!--\; -\; -->{12}^2}=13$
If $\text{If}z=x+yi\text{then}|z{|}^2=13\iff <!--\; \iff \; -->x^2+y^2=13$ then $\text{If}z=x+yi\text{then}|z{|}^2=13\iff <!--\; \iff \; -->x^2+y^2=13$

Solve the following system of linear equations by Gauss-Jordan elimination.
$$\begin{gathered} 3x_1+6x_2-<!--\; -\; -->9x_3=7 \\ 6x_1+3x_2-<!--\; -\; -->9x_3=6 \\ {x}_1+2x_2-<!--\; -\; -->3x_3=2 \end{gathered}$$

Fast way to come up with solutions to $x(x-<!--\; -\; -->1)(x-<!--\; -\; -->2)(x-<!--\; -\; -->3)=1$?

Solutions for a system of congruence equations
$$\{\begin{array}{l}x\equiv <!--\; \equiv \; -->7(\mathrm{mod}15)\\ x\equiv <!--\; \equiv \; -->14(\mathrm{mod}33)\end{array}$$

You have three numbers. The sum of these numbers are 7.2. The second number is twice as large as the first one. The third number is three times as large as the first one.
This is easily solveable by setting up an equation system such as:
$$a+b+c=7.2$$
$$b=2a$$
$$c=3a$$

Given the system of equations find $v_3,v_4$. If you only know the value of $v_1,v_2$
$p_0+p=p_1$
$p_1+p=p_2$
$p_0+2p=p_3$
$p_3+p=p_4$
$p_1{v}_1=p_2{v}_2=p_3{v}_3=p_4{v}_4$
Came to the equations when solving a complex physics task.

Solve for reals $x,y\in <!--\; \in \; -->\mathbb{R}$ given system of two non-linear equations.
$$\begin{array}{rl}5x(1+\frac{1}{x^2+y^2})& =12\\ 5y(1-<!--\; -\; -->\frac{1}{x^2+y^2})& =4\end{array}$$
I got
$$6x^{-<!--\; -\; -->1}+2y^{-<!--\; -\; -->1}=5$$
Substitute $x^{-<!--\; -\; -->1}=x_1$ and same for $y$ and got a four degree equation. Is there a short and elegant method for this?

Solve the following system of partial differential equations
$$\begin{array}{r}\{\begin{array}{l}\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}a}S(a,b,c,d)=f_1(a)\\ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}b}S(a,b,c,d)=f_2(b)\\ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}c}S(a,b,c,d)=f_3(c)\\ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}d}S(a,b,c,d)=f_4(d)\end{array}\end{array}$$
where $f_i()$s are some nonlinear functions.
Does the above system have a unique answer? And if has can any one introduce a reference, explaining the techniques for analytic solutions?