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OK, Solveit....although, technically.....Will beat me to it....!!!!!!

Thanks CPhill and Will85237

a(x - y)  = b(y - x)

ax - ay  = by - bx

ax + bx  = ay + by

x(a + b)  = y(a + b)

x / y  =  (a + b) / (a + b)    =  1

Apply algebra and play with it for a bit.

\(a(x-y)=b(y-x) \)

\(ax-ay=by-bx\)

\(ax+bx=by+ay\)

\(x(a+b)=y(a+b)\)

\(x=y\frac{a+b}{a+b}\)

\(x=y\)

Therefore,

\(\frac{x}{y}=1\)

3-(-3)=

3+3=

6

sec 2x = (sec^2 x)/ (2 - sec^2 x)

1 / cos2x  = (sec^2x) / (2 - sec^2x)  

Note, Shades....since    5/10   = 1/2    then it is also true that 10/5  = 2/1  .....thus....we can "flip" both fractions and write them as :

cos2x / 1  =  (2 - sec^2x) / (sec^2 x)  

cos2x =  2/sec^2x   -  sec^2x/ sec^2x

cos2x   = 2/ (1/cos^2x)  - 1

cos2x   = 2cos^2x  - 1        which is an identity   

-4(m+6) = -8

m+6 = 2

m = -4

 \(1500\times(1.03)^t=1635\\ t = 2.91547\)

Plug and chug buddy

1635 = 1500(1.03)^t

1635/1500 = 1.09

1.09 = 1.03^t

Log each side

log(1.09) = log(1.03)^t

Pull T out to front

log(1.09)=t*log(1.03)

Isolate T

t=log(1.09/1.03)

log(1.09)/log(1.03) = 2.9154664501233074

So about 2.92 units. Probably years in this instance.

I don't know what the trial and improvement is but i would just solve for x by factoring or the quadratic formula

I got x = 4.9

And 

X = -6.2

Let's suppose that there could be.....call the functions

f(x)  = ax + b      and  g(x)  = cx + d

So ......   a and c must be reciprocals and so must b and d....so we have

(ax + b) ((1/a)x + (1/b)  = 

x^2 + (b/a)x + (a/b)x + 1      

And this = x^2 + 1

Which implies that

[ (b/a) + (a/b)]    = 0

Which implies that

[b^2 + a^2] / ab   = 0

Which implies that

b^2 + a^2   = 0

But this is only possible if a and b  = 0

But....if a and b = 0 

Then f(x)  =  0x +  0   = 0

And   g(x)   = (1/0)x + (1/0)

Thus :   f(x) * g(x)   = 0 times an undefined expression   ....which could never be any function!!!

A shirt company has 3 designs each of which can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company?

So 3 designs ,short or long sleeves which is 2 and 7 colors:

\(3\times 2\times7=42\)

log₃(x)  = 1 / log₃(x)      multiply both sides by log₃(x)

[ log₃(x)]^2   = 1   take the square root of both sides

log₃(x)  = ± 1  

So, in exponential form we have that either :

3^-1  = x   = 1/3    ....or that......

3¹  = x  = 3

1412 factors as 2^2 x* 353   =   4 * 353

So

√1412  =

√[ 4 * 353 ]   =

√4 * √353  =

2√353

37.57658845611

A product originally costs $125. If the product goes on sale for 20% off, and then goes on sale for an additional 30% off, what is the price of the product ?

\(125\times0.8\times0.7=70\)

Angle  DWL = Angle CWK    [they are the same angle]
And angles  WLD  and WKC are both right angles   [all right angles are congruent ]

Then, by Angle-Angle congruency, ΔWKC  is similar to ΔWLD

Thus

DW      =    CW

LW             KW

No !

Let s look how much they want:

\(36\times0.25+36\times0.5+36\times0.75=54\)

Or 36 is 100% so let s look how many percent they want:

1/2 is 50%

0.75 is 75%

\(25\%+50\%+75\%=150\%\)

Thank you very much!

Sorry, Shades...I made this a little more complicated than I should have.....c is the focal distance  = sqrt(5)  = sqrt [ a^2 + b^2]   = sqrt [1^2 + 2^2] = sqrt [1 + 4 ]  = c

So,  sqrt [c^2 - b^2]  is  just = a   ......which, in this case = 1

So...I should have simplified the location of the vertices as

[ h + a , k ]   and [ h - a,  k]   where h = 1/3 and k = 3 and a = 1 

So....we get he same result as I gave originally

The vertices are located at

[1/3 + 1 , 3 ]   and  [ 1/3  - 1, 3]     which simpliy to

[4/3 , 3 ]   and   [ -2/3, 3 ]    as before

Sorry for the "overkill"

\(\angle {D}=\angle{C}\)

if a star has a parrallax of 0.026 second of arc and an apparent magnitude of 7, how far away is it and what is its absolute magnitude

If I remember my High school astronomy, you can calculate the distance of a nearby star if you know its parallax. Therefore:

D=1/p=1/.026 =38.46 parsecs.

38.46 x 3.26 =125.38 Light years-the distance to the star.

Not so sure about absolute magnitude, but I will give it a go!!.

Absolute magnitude is the brightness of a star at a standard distance of 10 parsecs. Since the distance to this star is 38.46 parsecs, then 38.46/10=3.846 times brighter. But since 1 magnitude represents a ratio of 2.512, then 3.846/2.512=1.5311. If we set the absolute magnitude to x, then we have: x^1.5311=7, and x=7^(1/1.5311)=3.564-The star absolute magnitude.

And that is the best I can do!!. Good luck.

Volume of the ice cream =  (4/3)pi *(2.5 / 2) ^3  = about 8.18 cu in

Volume of cone  = [ pi * (2.5/2)^2 * 4.75] / 3  = about 7.77 cu in

So....the cone cannot hold all the ice cream