geno3141

x - 2y  = 0     --->     x  =  2y

Substituting:  since  x + z  =  5   and   x  =  2y     --->     2y + z  =  5

Combining  2y + z  =  5   and   y - 2z  =  0

   2y + z  =  5   --->   x 2   --->   4y + 2z  =  10

    y - 2z  =  0                   --->     y -  2z  =    0

   Adding down:                        5y          =  10     --->     y  =  2

Since  y  =  2     --->     x  =  2y     --->     x  =  2(2)     --->     x  =  4

Since  x  =  4     --->     x + z  =  5     --->     4 + z  =  5     --->     z  =  1

Expanding  (a + bi)⁴  =  a⁴ + 4a³bi + 6a²b²i² + 4ab³i³ + b⁴i⁴  =  a⁴ + 4a³bi - 6a²b² - 4ab³i + b⁴

Since all the i-terms must cancel for the answer to be a real number:   4a³bi - 4ab³i - 24i  must have value 0.

   --->   4abi(a² - b²) - 24i  =  0

   --->   ab(a² - b²) - 6  =  0                 

   --->   ab(a² - b²) = 6 

Therefore, using factors of 6:  either  ab = 1  and a² - b² = 6  which is impossible because a = 1 and b = 1 and then a² - b² isn't 6

                                                      or  ab = 2  and a² - b² = 3  which means a = 2 and b = 1  -- this works!

                                                      or  ab = 3  and a² - b² = 2  which means a = 3 and b = 2 which is impossible

                                                      or  ab = 6  and  a² - b² = 1  which means a = 6 and b = 1 which is impossible

Now that you know that a = 2 and b = 1, place these back into the original formula   (a + bi)⁴ - 24i  to find the value of c.

Let A represent the total amount of money

      r represent the registration fee.

There were 124 new members. Since 50 had their initial registration fee waived, 74 paid the registration fee.

--->     Amount earned at registration:  A = 74·r

Assuming that no-one dropped out, 6 months later:

Let m represent the monthly fee

--->   Total amount earned for the first 6 months:  A  =  74·r + 124·m·6

Use these formulas:  m² + n²,  m² - n²,  2·m·n

For example:  If  m = 2  and  n = 1, then you have:  2² + 1²  =  5     2² - 1²​  =  3     2·2·1  =  4.

Must the point have integer coordinates?

If so, list all the possible points, count how many are not above the x-axis and count the total number of points, and divide the answers.

If not, there are an infinite number of points possible. The number of points on the x-axis can be can be added to those below the x-axis without increasing the (infinite) number. Then, there will be as many points below the x-axis as above the x-axis, so the answer will be 1/2. 

This isn't a geometric series ...

(1 - 2) + (3 - 4) + (5 - 6) + ... + (2005 - 2006) + 2007

   -1     +     -1   +     -1   + ... +          -1           + 2007

There are 1003 groups that result in - 1, so:   (1003)(-1) + 2007  =  -1003 + 2007  =  1004

The possible ways to toss coin B:                 The possible ways to toss coin A:

  H H                                                                          H H H                             H T T

  H T                                                                           H H T                             T H T

  T H                                                                           H T H                             T T H

  T T                                                                           T H H                              T T T

The probability of getting two Heads with coin B and three Heads with coin A:            (1/4)(1/8)  =  1/32.

The probability of getting one Head with coin B and two or three Heads with coin A:  (2/4)(4/8)  =  8/32.

The probability of getting no Heads with coin B and at least one Head with coin A:     (1/4)(7/8)  =  7/32.

Adding together these possibilities:  1/32  +  8/32  +  7/32  =  16/32  =  1/2.

arc(CD)  =  2 x angle(E)

arc(CD)  =  2 x 15

arc(CD)  =  30

arc(AB)  =  30

central angle(F)  =  arc(AB)  =  30

(AE) x (EB)  =  (DE) x (EF)

(3) x (x + 3)  =  (6)(x - 2)

3x + 9  =  6x - 12

       21  =  3x

         7  =  x

EB  =  x + 3  =  7 + 3  =  10

Since the equation for a circle is:  (x - h)² + (y - k)²  =  r²,

and h = 4 

and k = -3

and r = 2.5

the answer is:  (x - 4)² + (y - -3)²  =  2.5²     --->     (x - 4)² + (y + 3)²  =  6.25