Solve the linear system of equations.−5x −2y= 12x −11y=−83(x, y) =

Find the linear approximation of the function $f(x)=\sqrt{4-x}$ at $a=0$
Use L(x) to approximate the numbers $\sqrt{3.9}$ and $\sqrt{3.99}$ Round to four decimal places

Why is the equation ${\displaystyle \frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)}$ said to be standard form?
Well, I know that in linear differential equation the variable and its derivatives are raised to power of 1 or 0. But I am confused where did the standard form of linear differential equation came form?
That is, why is the equation ${\displaystyle \frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)}$ said to be standard form?

I have a system of linear equations of the form
${\displaystyle \sum _{j=1}^{n}2x_j{k}_{ij}\in \mathbb{Z},i=1,\dots ,m}$
or equivalently
${\displaystyle 2K\cdot \overrightarrow{x}\in {\mathbb{Z}}^n}$
where K is an integer matrix with orthogonal rows and columns, such that the GCD of every row and every column is 1, and ${\displaystyle \overrightarrow{x}}$ is a vector of real numbers.
what are the constraints on ${\displaystyle \overrightarrow{x}}$?

Find the augmented matrix for the following system of linear equations:
$\{\begin{array}{l}5x+7y-36z=38\\ -8x-11y+57z=-60\end{array}$

Given $m=385$, I have a linear equation system over a field ${\mathbb{F}}_p$ with $p$ a small prime (could be 5,7,11, something like this) with the following properties:
1.There are $m^2$ variables.
2. Every variable appears in exactly 7 equation.
3. Every equation contains at most 3 variables.
4. The coefficient of the variables is always 1.
5.The free term in the equations are always 0, with one specific equation (an equation of the form $3x=1$ for a specific variable $x$).
I am currently using SAGE. It solved nicely smaller equation systems, but this one killed it, even when constructing the matrix as sparse ("Error allocating matrix"). The question is - should I simply try a better (and less convenient) sparse matrix handling package, or is there a better way to deal with such sparse systems of equations? (I can do a little programming myself if needed).

I am trying to convert part of an equation from its log form into a linear form. Specifically, I am trying to convert ${\displaystyle {10}^{4\log \left(x\right)}}$, into ${\displaystyle x^4}$, but I'm really unsure of how to get from this first stage to the second. My experience with logarithms and exponents is limited, though I believe that ${\displaystyle {10}^{4\log \left(x\right)}}$ can be re-written as ${\displaystyle {10}^4+{10}^{\log \left(x\right)}}$, but I'm not sure that this helps my plight! Any very basic guidance would be greatly appreciated.

What I wanted to ask was , given a homogeneous system of n variables , like this having 4 variables:
${\displaystyle a_{1}x+a_{2}y+a_{3}z+a_{4}w=0}$
${\displaystyle a_{2}x+a_{3}y+a_{4}z+a_{1}w=0}$
${\displaystyle a_{3}x+a_{4}y+a_{1}z+a_{2}w=0}$
${\displaystyle a_{4}x+a_{1}y+a_{2}z+a_{3}w=0}$
Here , as we know that this system has zero solution and we can see that the coefficients are rotating in each of the linear equation .
So , is there any general form for solution of this system other than the zero solution ?
Also , what if we are given the non-zero solution then can we find the values of
${\displaystyle (a_1,a_2,a_3,a_4)}$