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x = y - 4

y = 4 - x

If you want a specific solution, like a number, there is not just one solution, for example..

x = 1     and     y = 5

or....

x = 0     and     y = 4

or....

x = -10     and     y = -6

Pick any number for x, and you can find one for y that will make the equation true (and vice versa).

So, there are infinitely many solutions to this.

The graph of all these solutions will be a straight line that will look like this:

I don't know what "z"  refers to, but...if it's the side length. we  just need to take the cube root of the volume....so we have....

∛4913  =   17 m

375¹⁶ = 152,932,674,524,208,778,166,212,141,513,824,462,890,625

Yep! That's right!

1 hour = 60 minutes        Multiply both sides of this equation by 12

12 hours = 720 minutes

Now we can set up this equation:

\(\frac{1500 \, \text{ papers}}{45\,\text{ mins}}=\frac{x}{720\,\text{ mins}}\)          and solve for x by multiplying both sides by 720 mins.

\(720\,\text{ mins}\,*\,\frac{1500 \, \text{ papers}}{45\,\text{ mins}}=x \\~\\ 24,000\,\text{ papers}=x\)

The bus driver would be y, with y standing for the age of the reader.

But what is y and what is x?

radius = diameter/2

radius = 6.5/2           =      \(\frac{13}{4}\)   m

volume  \(=\frac43 \, * \, \pi \, *\, (\text{radius})^3 \\~\\ =\frac43 \, * \, \pi \, *\, (\frac{13}{4})^3 \\~\\ =\frac43\,*\,\pi\,*\,\frac{2197}{64} \\~\\ =\frac{2197\pi}{48} \\~\\ \approx 144 \quad \text{cubic meters}\)

We can draw a line perpendicular to the top and bottom lines, which is the height, h

Two right triangles are formed on the ends of the trapezoid.

Since their hypotenuses are equal (5) , and one leg is equal (h) , the other leg must be equal (g) .

Since

g + 6 + g = 14          , we can solve for g .

2g + 6 = 14

2g = 8

g = 4

Now we can solve for h using the Pythagorean theorem.

g² + h² = 5²

4² + h² = 5²

16 + h² = 25

h² = 9                    take the + square root of both sides

h = 3

Now we know the height of the trapezoid and the length of both bases.

area of trapezoid = [ (6 + 14)/2 ] * 3

area of trapezoid = [ 10 ] * 3

area of trapezoid = 30     sq. un.

5x - 5y = -20

We can divide through by 5, which gives us

x - y = -4

What you mentioned is only a fraction of that story:

This story is called "hilbert's hotel" and it teaches, in a very sophisticated way, about cardinal number of different sets, and compares them to different cardinal numbers of different groups (talks only about infinite cardinal numbers, mostly about the smallest infinite one, aleph 0, the cardinal number of the set containing the natural numbers: 1,2,3,4...... also called "the countable infinity")

And again, before you say that "infinity plus/times something else is still infinity" you have to translate it to the "set theory" language and CLARIFY THE MEANING. otherwise, it means nothing.

\(\angle{ABC}=\frac12\,\cdot\,\overset{ \frown}{AC} \\~\\ 25^{\circ}=\frac12\,\cdot\,\overset{\frown}{AC} \qquad \text{ Multiply both sides of the equation by 2.} \\~\\ 50^{\circ}=\overset{\frown}{AC}\)

Let theta be the angle the stream makes with the road.

\(\cos \theta = \dfrac{20}{27}\\ \theta = \cos^{-1}\dfrac{20}{27}\approx 42^{\circ}\)

This stream makes a 42-degree-angle with the road.

MAX: JUST SHEER BRILLIANCE !! CONGRATS.

Anyone heard of this story?

There is an apartment where there are infinite numbers of rooms(and those are full for some reason...) and one person more wants to live here too, so everyone move to the next room(The guy who lives in room 1 moves to room 2, the guy who lives in room 2 moves to room 3,.... etc.). So that person also have a room! :)

And then there are a bunch of people whose number is same as the number of residents, they all wanted to live in the apartment. Then everyone move to the room of (number of original room * 2) (The guy who lives in room 2 moves to room 4, the guy who lives in room 3 moves to room 6... etc). So all people have a room :)

Conclusion: Infinity is awesome :)

My rules of dealing with infinity:

1) ∞ + C = ∞, where C is any number.

2) C * ∞ = ∞ , same as above

3) ∞^C = ∞, same as above

Because there are nothing bigger than infinity, adding something to infinity is still infinity. Imagine a wall is blocking your way (because no bigger than infinity)and there are someone pushing behind you(the addition/multiplication/exponent), and that guy pushing behind you won't make you penetrate through the wall...XD

So that's why (I think) the 3 rules are true.

1/2 x 12 x 9 = 6 x 9 = 54 :D

5? :D

Hmm Idk

I only know that the square of root beer is just beer... XD

About 524 years old?

That's what I know... XD

Use ur calculator :D

118,863,898

\(\dfrac{2\sqrt[3]{9}}{1+\sqrt[3]3+\sqrt[3]9}\\ \boxed{\text{Remember that }a^3 - b^3 = (a - b)(a^2+ab+b^2) }\\ =\dfrac{2\sqrt[3]9}{1+\sqrt[3]3+(\sqrt[3]{3})^2}\\ =\dfrac{(2\sqrt[3]9)(1-\sqrt[3]{3})}{(1-\sqrt[3]{3})(1+\sqrt[3]3+(\sqrt[3]{3})^2)}\\ =\dfrac{(2\sqrt[3]9)(1-\sqrt[3]{3})}{1^3-(\sqrt[3]{3})^3}\\ =\dfrac{(2\sqrt[3]9)(\sqrt[3]{3}-1)}{2}\\ =\sqrt[3]{9}\times(\sqrt[3]{3}-1)\\ =\sqrt[3]{27}-\sqrt[3]{9}\\ =3 - \sqrt[3]{9}\\ =3 - 3^{2/3}\)

Exactly what CPhill said!! :D

Yea. :D 

I am happy to teach you some math... :P 

(I am only 14 and I know some calculus)

What exactly do you want to learn? ;) 

Press the "2nd" button and find the "asin" button. It does the same thing as sin-1

\(a_1=\dfrac{2}{1\cdot 2 \cdot 3}=\dfrac{1}{3}\\ a_1+a_2 = \dfrac{2}{1\cdot 2\cdot 3} + \dfrac{2}{2\cdot 3\cdot 4}=\dfrac{5}{12}\\ a_1+a_2+a_3=\dfrac{2}{1\cdot 2\cdot 3} + \dfrac{2}{2\cdot 3\cdot 4}+\dfrac{2}{3\cdot 4\cdot 5}=\dfrac{9}{20}\\ a_1 + a_2 + a_3 + a_4 = \dfrac{2}{1\cdot 2\cdot 3} + \dfrac{2}{2\cdot 3\cdot 4}+\dfrac{2}{3\cdot 4\cdot 5}+ \dfrac{2}{4\cdot 5\cdot 6}=\dfrac{7}{15}\\ a_1 + a_2 + ... + a_5 = \dfrac{7}{15}+\dfrac{2}{5\cdot 6 \cdot 7}=\dfrac{10}{21}\\ a_1 + a_2 + ... + a_6 = \dfrac{10}{21} + \dfrac{2}{6\cdot 7 \cdot 8}=\dfrac{27}{56}\)

It seems like it will not exceed 1/2.

So the answer is 1/2