Alan

$$\frac{4}{7}*bill=200\\\\
bill=200*\frac{7}{4}$$

$${\mathtt{bill}} = {\frac{{\mathtt{200}}{\mathtt{\,\times\,}}{\mathtt{7}}}{{\mathtt{4}}}} \Rightarrow {\mathtt{bill}} = {\mathtt{350}}$$

bill = $350

$$6x^2-8x=6x^2-24x-18\\
-8x=-24x-18\\
16x=-18\\
x=\frac{-18}{16}\\
x=-\frac{9}{8}$$

Edit:  Corrected version. 

On the calculator here, press Enter or = twice.

For example if you do 3 + 1/2 and press = you get 3.5, but if you press = again you get 7/2.

In math it usually means that when a value is put into a function, there is only one possible output. One value in produces one value out.

For example, think of the function y(x) = sin(x).  Whatever value of x we choose there is just one corresponding value of y.  If x = 45°, say, then y(45°) = 1/√2. One value in, one value out.

However, if we had the function y(x) = sin^-1(x),  then if x is 1/√2, there is more than one possible value for y - it could be 45° or 135° for example. One value in, more than one out.

It can take any value in the range 0 to +1 - see graph (note that the fourth quadrant is between 270° and 360°):

Assuming the temperature is the same (which it won't be in reality!) then pressure*volume is constant, so:

80.5*1 = V*0.55  (Taking pressure at sea level as 1 atm)

V = 80.5/0.55 L

$${\mathtt{V}} = {\frac{{\mathtt{80.5}}}{{\mathtt{0.55}}}} \Rightarrow {\mathtt{V}} = {\mathtt{146.363\: \!636\: \!363\: \!636\: \!363\: \!6}}$$  L

V ≈ 146.4L

The vectors are in the third quadrant and their angles with respect to the horizontal axis are obtained from tan(angle)=y_value/x_value, so the angle between them is given by:

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\frac{\left(-{\mathtt{4}}\right)}{\left(-{\mathtt{5}}\right)}}\right)}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\frac{\left(-{\mathtt{3}}\right)}{\left(-{\mathtt{4}}\right)}}\right)} = {\mathtt{1.789\: \!910\: \!608\: \!246}}$$ degrees

or 1.8° to the nearest tenth of a degree.

e^ln(x) = x       "ln" means log to the base e

Why not just divide every term by 5?  The answer is immediate:  x-2 = 5

Hmm!  Unusual to have 180 radians.  What if the 180 is 180°, or $$\pi$$ radians? In this case we get a different set of results:

The values of x are now given by $$x=(n+\frac{1}{2})\pi$$ where n is an integer.