ConfuzzledKitten

Alright, thanks!  I will try what you said :)

I didn't quite get what geno said, and I tried something different.  I basically used law of cosines to find all 3 side lengths of ABC in terms of AE, then solved for it with pythagoras.  It worked!

I tried what you said up until the last step, I'm probably missing ONE tiny bit but I can't seem to figure out how to do it...

Thanks!  I will try and do that! :) :)

Thank you!! I got it!

Could you explain to me how you got that answer?

I think the image you linked is local to your computer, so we can't see it...

Thank you!  I searched up a proof on Google and I think I understand now! :-)

Thank you for replying!  I do not really know how you got the first equation, I might be missing something really simple, but if you have the time, could you explain it to me? Thanks!

You can re-write the expression to be ((2a-3b)^2)^2.  First, you can simplify the inside by using the square of binomials formula: (a+b)^2=a^2+2ab+b^2

This gives 4a^2-12ab+9b^2.  Then you can multiply each term, and get a really big number: 16a^4 - 96a^3*b + 216a^2*b^2 - 216ab^3 + 81b^4

Hi! Thank you guys for responding.  I solved it!  I did it like this:

The base 7 number can be "abc_7", and a, b, and c are all digits from 1 to 6, except for b which can also be 0.  Then you can expand it to be 49a + 7b + c.  Then when it is reversed, you get cba.  expanding in base 9, cba_9= 81c+ 9 b + a.  The two expressionss are equal, so you can use the equation 49a + 7b + c = 81c + 9b + a.  Then you move everything to one side and get 80c + 2b - 48a = 0, and you can simplify and get 40 c+ b- 24a = 0.  Then we can solve for b and get  b   = 24a - 40c = 8(3a - 5c). This means that b is divisible by 8, but it is also a base 7 digit so it has to be 0.  then you have 3 a = 5c sp a is divisible by 5 and c is divisible by 3.  They cant be 0 because they are both left digits, so they are 5 and 3.  This means that the answer is 503_7.

Hooray!  Thank you again for helping me :)

Subtract 3y from the first equation to get x = -3y - 4

Substitute x into the second equation to get -9y - 12 - 7y = 36

Simplify: -16y - 12 = 36   -->   -16y = 48   -->   y = -3

Solve for x using the first equation: x = 9 - 4 = 5

Your answer is ( 5 , -3 )

Hello! I am new to this website and I am not the best at math, but I like helping people so I will try to help.  Please tell me if I get it wrong, and don't be mad. Thanks!

My thinking is that there are exactly five possible even digits, and we can first take a look at the possible 4-digit numbers where the first digit is even.  There are 4 choices for the first one because it can't be 0, and 10 for each one following.  This gives 4*10*10*10=4000.  We can do the same for when the even digit is in the hundereds, tens, and ones place, then add them all up: 4000+5000+5000+5000=19000.  So the answer is 19000.  Again, I might not be correct. :)