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No....

They either intersect at one point  (one solution)  or

They are parallel  (no intersections at all....so...no solutions )  or

They are the same line and intersect at an infinite number of points

This might help  :  http://nogales.laurusmath.com/cpages/contents/pdf/HS-S3-C3-PO4-notes.pdf

https://www.symbolab.com/solver/equation-calculator/%5Csin(2x)+%5Csin(x)=0,0%3C=x%3C=360

I can try! Message me? 

multiply both by lw

l²+w²=l²​^w

subtract l²

w²=w

subtract w

w=0

??

Total cost of party in 2007  = p*d

Total cost of party in 2008  = (2p) (2d)  = 4 pd

So....2007 cost   / 2008 cost  =  pd  / 4pd  =  1/4

2007d = 1/2 2008 d

2007p = 1/2 2008 p

simple, 1/2

2/5  =  12/30

1/2  = 15/30

1/3  = 10/30.....so

28 2/5  -  5 1/2  =

28 12/30  - 5 15/30  =

27 42/30 - 5 15/30  =

22 27/30

And

22 27/30  - 7 10/30  =

15 17/30   apples left

f(x) + g(x)  will also be quadratic functions opening dowward.....we cannot make any general statement about the max of the new function formed

For example  if  f(x) = -2x^2, g(x)  = -3x^2  then f(x) + g(x)  = -5x^2

And all of these have the same max at the origin

But  f(x) = -2x^2 + 3  and  g(x) = -3x^2 + 5    have maxes at (0, 3) and (0, 5), respectively

But f(x) + g(x)  =  -5x^2 + 8  has a max at (0, 8)

Let the number of nosebleed tickets sold  = T.....the the number of mezzanine tickets sold  = (21500 - T)

So

Number of nosebleed tickets sold * price each + number of mezz tickets sold * price each = revenue

T(20)  + ( 21500 - T)(45)  =  717500      simplify

20T + 967500 - 45T  =  717500

-25T + 967500 = 717500       subtract  967500 from each side

-25T  = -250000     divide both sides by -25

T = 10000  = nosebleed tickets sold   and    21500 - 10000  = 11500 mezz sold

So.....

Revenue from mezz =  45(11500)   =  $ 517500

Revenue from nosebleed  =  717500 - 517500  = $ 200000

So....they made   517500 - 200000  = $317500  more from mezz sales than nosebleed sales

25x^2 + 36        this is a sum of squares.....it cannot be factored in "normal" terms.....however....it can be factored  like this :

(5x + 6i)  ( 5x - 6i)

2000

srry but no it is 5

N mod 7 = 2
N mod 9 = 4
N mod 12 = 7, solve for N

The LCM[7, 9, 12] =252
N =252n + 247, where n =0, 1, 2, 3.....etc.
Sum=247 + 499 + 751 = 1,497

333^333 mod 33 = 27

333^1 mod 33 = 3

333^2 mod 33 = 9

333^3 mod 33 =27

.

.

.

333^8 mod 33 =27

333^333 mod 33 = 27

That is pretty ingenious and is better than what I did!

Based on my past answer of 54 (you posted the same question earlier and I incorrectly answered 54), I noticed that there are only 2 cases that don't fit.

So 54 - 2 = 52

So 52 is the answer?

Thanks! I wasn't sure if Modulo 7^2019 worked on this kind of problem, and yeah, I think it does.

So my way (that I didn't know that worked) was to see the ones digit for powers of the remainder of 7 mod 5.

2, 4, 8, 6

It would be 8, and then divide that by 5 and get the remainder of 3.

Thanks!

1. Simple pattern recognizing.

Remainder of when divided by 5- 

7^1=2

7^2=4

7^3=3

7^4=1

7^5=2

and so on

This means the remainder pattern is 2, 4, 3, 1,

To find the remainder of 7^2019, we find the remainder of 2019 divided by 4 because there are 4 terms in the pattern.

Since the remainder is 3, we count up 3 times in the pattern.

So the remainder of 7^2019 divided by 5 is 3.

x can represent the total gallons of gasoline.

This means that when it was \(\frac{1}{8}\)full, it is \(\frac{1}{8}\)x.

So this means \(\frac{1}{8}x+15=\frac{3}{4}x\)

Solving we get 24 gallons in the entire tank.

It is currently three-quarters full, which is 18 gallons out of a total of 24 gallons.

This means we need \(\boxed{6}\) more gallons to fill the tank.

I posted another question on y profile called " Pleas explain how they got these values" ,I am very confused.

Imaginary number rules,

To the power of -

1: i

2: -1

3: -i

4: 1

5: i

and so on.

-------------------------------

Now we have to find how many times each will occur throughout the sequence.

Since 96th term will be 1, this means that each power of i outcome will occur 24 times. So we have 24i-24-24i+24i

97th term will be i

98th term will be -1

99th term will be -i

So add the three above to the expression we had earlier to get : 25i-25-25i+24i

Simplifies to -25+24i

\(\boxed{-25+24i}\)