5y+3=2y-3x+5y = ___

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

If a system of linear equations has infinitely many solutions, then the system is called _____. If a system of linear equations has no solution, then the system is called _____.

If we were given two points on a linear equation ${\displaystyle (x_1,y_1),(x_2,y_2)}$, it is quite easy to find the slope and use substitution to find the slope intercept form ${\displaystyle y=mx+b}$, to graph it.
Is it possible to solve for b strictly in terms of ${\displaystyle x_1,y_1,x_2,y_2}$?

Solve (1,2)

$$\begin{gathered} 3x-y=1 \\ 2x+3y=8 \end{gathered}$$

Consider the following linear system of differential equations:
$\{\begin{array}{l}\dot{x}=-4y\\ \dot{y}=x\end{array}$
where $x(t)$ and $y(t)$ are unknown real functions.
One can simply verify that the general solution is
$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)=c_1\left(\begin{array}{c}\cos (2t)\\ \frac{1}{2}\sin (2t)\end{array}\right)+c_2\left(\begin{array}{c}-2\sin (2t)\\ \cos (2t)\end{array}\right)$
where $c_1$ and $c_2$ are real parameters.
Question: Which is the exponential form of this expression? It should by something like
$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)=k_1{e}^{i2t}\left(\begin{array}{c}{P}_{11}\\ P_{21}\end{array}\right)+k_2{e}^{-i2t}\left(\begin{array}{c}{P}_{12}\\ P_{22}\end{array}\right)$
where $k_1$ and $k_2$ should (?) be complex parameters.

Write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and (b) perpendicular to the given line. Point (−1, 0) Line y=-3

How to solve the system of linear differential equation of the form
${\displaystyle x^{\prime }=Ax+b}$
I can solve the homogeneous form by finding the eigenvalues and respective eigenvectors, but how to find the particular solution part. Also is there any limitation from getting eigenvalues positive, negative or complex. Any other different method involving matrix algebra is also welcome.