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I think i know how to prove it im in 2nd year in Greek university in Math science but this is a lesson for 1st year 

and I'm doing it again!

I know that, but I'm not going to give you the answer, I want you to try first

The system will have infinite solutions when both equations are  exactly the same

Note that we can add 4y to both sides of the second equation and get

6x + 8y   =  c

Notice that the left side is just a multiple  [ of 2 ] of the left side of the  first equation

So...multiplying the right side of the first equation by 2  gives us 14  = c  and both equations will be exactly the same.....so....the system will have infinite solutions!!!!

First you must prove that,A is invertible because if it don't we can't prove Α^-1 = -Α^4

You know the question better than me, are you supposed to assume that A is invertible? It IS possible to prove that A is invertible, so I don't think you're supposed to assume that.

If it's not we can't prove Α^-1 = -Α^4 

Ohh Thank you very much Alan!Have a nice day!

You assumed that A is an invertible matrix

"If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4"

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I would not call it streched as it is being shrunk but anyway.

\((3x)^2-y^2=1\\ is\\ 9x^2-y^2=1\)

So x is replaced  by 3x just as you wanted to do :)

And for b we do not know anything?

3) the point A has coordinates (-3,2) and the point B has coordinates (7,k)

the line AB has equation 3x+5=1

a) i) show that k= -4

ii) hence find the coordinates of the midpoint AB

b) find the gradient of AB 

c) a line which passes through the point A is perpendicular to the AB. Find an equation of this line, giving your answer in the form px + qy + r = 0. Where p,q and r are intergers

d) the line AB, with equation 3x+5y=1 , intersects the line 5x + 8y = 4 at the point C. Find the coordinates of C

Charming, guest  :)

LOL

A “do-over” power is a great idea! I would have used it after my date with Mila.

Mila was a very popular girl in my high school, but she rarely went out with anyone. For whatever reason, she went on a date with me. The date went well, and she even kissed me goodnight. I thought it would be great to slip her the tongue. She smacked the heII out of me and never again would go out with me. Having a “do over” power would have been great. I would have waited until the second or third date to slip her the tongue. 

This super power might have caused me more problems in the long run, though. I didn’t know it then, but Mila had “do over” powers: she married four times in the eighteen years following high school. Number four was her last marriage but not because it was the charm. Her husband took the “death do us part” vow very seriously and strangled her three weeks after vowing to love, honor and cherish.

Mila lost her “do over” powers after that, so she never remarried, and she’s still dead.

Looking back, I’m really glad I never had them. 

\(\text{we need the lines to have the same slope and intercept}\\ 3x+4y=7 \Rightarrow y = -\dfrac 3 4 x + \dfrac 7 4\\ 6x+4y=c-4y \Rightarrow y= -\dfrac 3 4 x + \dfrac c 8\\ \dfrac c 8 = \dfrac 7 4 \\ c = 14 \)

Notice that we have four corner rectangles in the darker strip, each with dimensions of  2 x 1  = 2 unit^2

So....the total area of these  = 8 units^2

The other parts of the dark gray strip would be rectangles with the same dimensions as the rectangles  comprising the area of the lighter gray strip...so...their total areas  =  100

So...the  area of the darker gray strip  = 100 + 8  =  108 sq units

 Corrected answer  !!!

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I'm sorry that this is so late, but thank you so much for taking the time to help me on this question!

I'm very sorry Melody! Even though you took time out of your day to help me with the problem multiple times, I completely failed to thank you. I also never apologized to you for reposting the question without properly apologizing that I wasted everyone's time on a problem flawed by my own carelessness. Once again, I am so sorry and thank you for being patient with me, despite all of my wrongdoings!

I deeply apologize for all of the trouble I have caused and thank you so much for helping me even though I have behaved so poorly on my part!

100 integers from 500 - 599 contain at least one 5

And in each of the 499 integers

_5, _15, _25, _35, _ 45, _55 , _65, _ 75, _ 85 , _ 95

In each hundred  from 1 to 499 contain at least one 5

And there are 5 sets of these = 5 * 10  =  50

And in each hundred from 1 to 499 we have the  integers

_5_   where the last digit is not a 5

And we have 9 of these in every hundred

So...we have  5 * 9  = 45 of these

So....100 + 50 + 45  = 195

Just as  Dimitristhym  found  !!!

Thank you so much for helping me!

Thank you so much!

Dear Melody, Hectitar, and all of the Guests who helped me on my problems,

Hello, I am Guest #1. I am extremely sorry for giving everyone a flawed question, and even more so, for not thanking anyone for their efforts. Even though you guys used so much of your time to try and help me solve the problem, I have completely failed in saying thank you to everyone in the original post. I have also failed to properly apologize in this reposting of the question for my error. I am truly sorry and I will make sure to never be this ungrateful and rude again. Thank you and sorry to all of you, who have been so patient even though I have demonstrated such a poor attitude. 

Extremely sorry,

Guest #1