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 _____.

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

Write the equation of the line that passes through the point (-4, -9) and has a slope of 12

find an equation of the form $y=a{x}^2+bx+c$, whose graph passess through the points (2, 3), (-2, 7) and (1, -2). You must use a system of linear equations in three variables for this problem.

Suppose the 4-vector c gives the coefficients of a cubic polynomial ${\displaystyle p\left(x\right)=c_1+c_{2}x+c_3{c}^2+c_4{c}^3}$. Express the conditions
${\displaystyle p\left(1\right)=p\left(2\right),p^{\prime }\left(1\right)=p^{\prime }\left(2\right)}$
as a set of linear equations of the form Ac=b. Give the sizes of A and b, as well as their entries.
So far, what I've done was sub in the values and differentiate the polynomials and let it equal to each other after subbing their values.
I get
${\displaystyle c_1+c_2+c_3+c_4=c_1+2c_2+4c_3+8c_4}$
then,
${\displaystyle 0=c_2+3c_3+7c_4}$
as well as
${\displaystyle c_2+2c_3+3c_4=c_2+4c_3+12c_4}$
then,
${\displaystyle 0=2c_3+9c_4}$
What should I do now?

Let A be an abelian group, ${\displaystyle a_i}$ variables indexed with some arbitrary set I and assume we have an infinite set E of linear equations in finitely many variables of the form
${\displaystyle n_1{a}_1+\dots +n_k{a}_k=b}$
with ${\displaystyle n_i\in \mathbb{Z},b\in A}$.
If this system has no solution, then is there already a finite subset of equations in E which are inconsistent? So if any finite subset of equations is solvable, is E solvable?
This is reminiscent of the compactness theorem in 1st order logic, but of course you can't apply this for a concrete group. Am I missing something? I assume that this has an easy proof or a counterexample. What happens if we presume particularly nice groups (divisible, free) or vector spaces? Then this question is just about linear systems of equations.

Write the augmented matrix for the system of linear equations
2y-z=7
x+2y+z=17
2x-3y+2z=-1

a) Determine a condition under which (x, y, z) is a linear combination of [-3, 5, -3], [-9, 11, -3], [-6, 8, -3]? Your condition should take the form of a linear equation.
Im so confused?? I have the theorem that every vector (x, y, z) in ${\displaystyle R^3}$ is a linear combination of ${\displaystyle x(1,0,0)+y(0,1,0)+z(0,0,1)}$.