The standard form of a linear firs-order DE is d y </mrow> d

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

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y. or x, y, z. or $x_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $\left[\begin{array}{cccc}1& 0& 4& 4\\ 0& 1& 3& 2\\ 0& 0& 0& 0\end{array}\right]$

Let ${\displaystyle A\in M_{m\times n}\left(\mathbb{Q}\right)}$ and ${\displaystyle b\in {\mathbb{Q}}^m}$. Suppose that the system of linear equations Ax=b has a solution in ${\displaystyle {\mathbb{R}}^n}$. Does it necessarily have a solution in ${\displaystyle {\mathbb{Q}}^n}$?
and I thought I'd give an interesting, possibly wrong, approach to solving it. I'm not sure if such things can be done, if not maybe you can help me refine.
I considered the form of the equality as
${\displaystyle A^{\left(1\right)}{x}_1+\cdots +A^{\left(n\right)}{x}_n=b}$
where ${\displaystyle A^{\left(i\right)}}$ is a column vector of A. I then noticed that for ${\displaystyle x_i\in \mathbb{R}\mathrm{\setminus }\mathbb{Q}=\mathbb{T}}$ then, and this is where I think I'm doing something forbidden, each x has the represenation
${\displaystyle x_1=k_{11}{\tau }_1+\cdots +k_{1p}{\tau }_p}$
${\displaystyle x_2=k_{21}{\tau }_1+\cdots +k_{2p}{\tau }_p}$
${\displaystyle ⋮}$
${\displaystyle x_n=k_{n1}{\tau }_1+\cdots +k_{np}{\tau }_p}$,
where ${\displaystyle {\tau }_i}$ is a distinct irrational number, ${\displaystyle k_{ij}\in \mathbb{R}}$, and p is the number of such distinct irrational numbers. I wound this out, but there may be a discrepancy with p and m. I feel this method can lead me to the answer, but I'm not sure where to go from here.
I end up getting something like this, I believe, after substitution:
${\displaystyle A(k^{\left(1\right)}{\tau }_1+\cdots +k^{\left(p\right)}{\tau }_p)=b}$
Here, ${\displaystyle k^{\left(i\right)}}$ is the vector
$((k_{1i}),(\dots ),(k_{ni}))$.
I think there is no discrepancy with p and m because ${\displaystyle A\in M_{m\times n}\left(\mathbb{Q}\right),K\in M_{n\times p}\left(\mathbb{R}\right)}$, and ${\displaystyle \tau \in M_{p\times 1}\left(\mathbb{T}\right)}$, so
${\displaystyle (m\times n)\cdot (n\times p)\cdot (p\times 1)=m\times 1}$.

Find the slope of the line and simplify the answer.
-9x = 2

My book says the general form of a linear differential equation is:
${\displaystyle y^{\prime }+P\left(x\right)y=Q\left(x\right)}$
But my teacher said that a linear differential equation has the general form:
${\displaystyle y{}^{″}+P\left(x\right)y^{\prime }+Q\left(x\right)y=f\left(x\right)}$
Which is the correct form or they are both correct?

Solve the linear system of equations.
$-5x-2y=1$
$2x-11y=-83$
$(x,y)=$

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. All vectors (x, y, z) in $V_3$ whose components satisfy a system of three linear equations of the form:
$a_{11}x+a_{12}y+a_{13}z=0$
$a_{21}x+a_{22}y+a_{23}z=0$
$a_{31}x+a_{32}y+a_{33}z=0$