1. ## Aligning symbols

Code:
\begin{array}{r@{.}l}
12341 & 43 \\
& 99 \\
23    & 12 \\
10    & \\
938   & 01
\end{array}
$
\begin{array}{r@{.}l}1
2341 & 43\\
& 99\\
23&12\\
10 & \\
938 & 01
\end{array}
$

Code:
\left\{
\begin{array}{r@{+}r@{+}r@{=}l}
u   & (-v) & w       & -1 \\
3u  &      & (-9)w   & 47 \\
-2u & 5v   & 3w      & 31
\end{array}
\right.
$\left\{
\begin{array}{r@{+}r@{+}r@{=}l}
u & (-v) & w & -1\\
3u & & (-9)w & 47\\
-2u & 5v & 3w & 31
\end{array}
\right.$

Code:
\left\{
\begin{array}{r@{\ +\ }r@{\ +\ }r@{\ =\ }l}
u   & (-v) & w       & -100 \\
3u  &      & (-9)w   & 47 \\
-2u & 5v   & 3w      & 31
\end{array}
\right.
$\left\{
\begin{array}{r@{\ +\ }r@{\ +\ }r@{\ =\ }l}
u & (-v) & w & -100\\
3u & & (-9)w & 47\\
-2u & 5v & 3w & 31
\end{array}
\right.$

2. I prefer:

$$\left\{\begin{array}{rcrcrcc} u &- &v&+ & w & =&-100\\ 3u & & & -&9w &=& 47\\ -2u &+& 5v &+& 3w &=& 31 \end{array} \right.$$

$\left\{\begin{array}{rcrcrcc}
u &- &v&+ & w & =&-100\\
3u & & &- &9w &=& 47\\
-2u &+& 5v &+& 3w &=& 31
\end{array}
\right.$

(less control of the spacing, but you can have whatever signs you want)

(Note use of [noparse][/noparse] tags to show LaTeX source, and to show the tags for that matter)

.

3. Another method is to use the aligned environment, using \phantom to get spaces where required.

\begin{aligned} u-\phantom{5}v+\phantom{9}w &= -100 \\ 3u \phantom{{}+5v} -9w &= \phantom{-1}47 \\ -2u+5v+3w &= \phantom{-1}31 \end{aligned}

\begin{aligned}
u-\phantom{5}v+\phantom{9}w &= -100 \\
3u \phantom{{}+5v} -9w &= \phantom{-1}47 \\
-2u+5v+3w &= \phantom{-1}31
\end{aligned}

4. It's a bit of extra work, but I use \text{-} to shorten the minus signs.

First I type:

\begin{pmatrix}1 & -2 \\ -3 &1 \end{pmatrix}

. . . $\begin{pmatrix}1 & -2 \\ -3 &1 \end{pmatrix}$

I preview the result, then I insert the "\text{ }"

\begin{pmatrix}1 & \text{-}2 \\ \text{-}3 &1 \end{pmatrix}[/tex]

. . . $\begin{pmatrix}1 & \text{-}2 \\ \text{-}3 &1 \end{pmatrix}$

I use it whenever I have negative quantity, rather than a subtraction:

. . . $f(\text{-}1) \;=\;(\text{-}1)^2 - 3(\text{-}1) + 4 \:=\:8$

5. $\mathbb{H}^n=\left\{\left(\begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n\end{array}\right)\in\mathbb{R}^n:x_n\geqslant 0\right\}$

$\partial\mathbb{H}^n\subset\mathbb{H}^n$

$\partial\mathbb{H}^n=\left\{\left(\begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n\end{array}\right)\in\mathbb{R}^n:x_n=0\right\}$

6. $\Downarrow$ $\downarrow$ $\Uparrow$ $\uparrow$ $\leadsto$ $\swarrow$ $\leftrightsquigarrow$

7. it appears based on gravitational attractions there is going to be two tides per day

now consider this

height $\rightarrow \frac{atidal}{g}
$

$atidal=\Delta g_{moon}=GM_m \frac{1}{(D_{em}+R_e)^2} - \frac{1}{D^2_{em}}=-\frac{2GM_mR_e}{D^3_{em}}=10^{-7}g$

when $D_{em}=3.84 * 10^8 m$

this produces tides about 1m high taking into consideration

$R_e=6.378 * 10^6$

8. How do you mix text and mathematical expressions withing a single [tex] "block" ?

9. Originally Posted by Jones
How do you mix text and mathematical expressions withing a single [tex] "block" ?
Text can be embedded in displayed equations (in LaTeX) by using \mbox{embedded text}. For example, one obtains

$M^\bot = \{ f \in V' : f(m) = 0 \mbox{ for all } m \in M \}$
by typing
M^\bot = \{ f \in V' : f(m) = 0 \mbox{ for all } m \in M \}

Another example:

$\int_0^{2\pi}\cos(mx)\,dx = 0 \hspace{1cm}\mbox{if and only if} \hspace{1cm} m\ne 0$

by typing:

\int_0^{2\pi}\cos(mx)\,dx = 0 \hspace{1cm}
\mbox{if and only if} \hspace{1cm} m\ne 0

10. $f(x)=\left\{\begin{array}{cc}\frac{e^x -e^a}{x-a},&\mbox{ if }
x\ne a\\K, & \mbox{ if } x=a\end{array}\right.$

11. $\limsup$

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