x + 2y = -1 → x = -1 -2y (1)
3y - x = -19 + x (2)
Sub (1) into (2)
3y - (-1 -2y) = -19 + (-1 -2y)
3y + 1 + 2y = -19 -1 -2y
5y + 1 = -20 - 2y
7y = -21
y = -3 and x = (-1 -2(-3)) = 5
xy = 5(-3) = -15
(2^10)^1 mod 18 = 16 (3^10)^1 mod 18 = 9
(2^10)^4 mod 18 = 16 (3^10)^2 mod 18 = 9
(2^10)^(3n - 2) mod 18 = 16 (3^10)^(n) mod 18 = 9
So
(2^10)^7 mod 18 = 16 (3^10)^7 mod 18 = 9
((2^10)^7 + (3^10)^7) mod 18 =
(16 + 9) mod 18
(25 mod 18) = 7
1/ ( a^3 + 7) - 7 = -a^3 / ( a^3 + 7) rearrange as
1 / (a^3 + 7) + a^3/ (a^3 + 7) = 7
(1 + a^3) = 7 ( a^3 + 7)
1 + a^3 7a^3 + 49
-6a^3 = 48
a^3 = -48 / 6
a^3 = -8
a = -2
2. -3cos ( pi - 2x) = 1
Note
cos (pi -2x) = cos (pi) cos (2x) + sin (pi) * sin (2x) = -1cos (2x) + 0* sin (2x) = -cos (2x)
-3 (-cos (2x) ) = 1
3 cos (2x) = 1
cos (2x) = 1/3
arccos (1/3) = x ≈ .123 rads
2x = 1.23
x = 1.23 / 2 ≈ .615 rads
So x = [ .615 + 2pi * n ] rads where n is an integer
Also
x = [ (2pi - .615 ) + 2pi*n] rads ≈ [ 5.668 + 2pi*n ] rads where n is an integer
1. Note that sin ( x - pi) = sin x cos (pi ) - sin( pi ) cos x = sinx * (-1) - 0 * cosx = -sin x
2sin ( x -pi ) - 1 = 0
2 (-sin x) - 1 = 0
-2sin x = 1
sin x = -1/2
x = 7pi / 6 + n* 2pi and x = 11pi/6 + n* 2pi
Where n is an integer
1 plays only an instrument
8 play a sport and and instrument
9 play only a sport
5 do neither
Sport No Sport Total
Instrument 8 1 9
No Instrument 9 5 14
_______________________
17 6 23
P ( no instrument given a sport) = 9 / 17
19t + 5/t + 2/t =
[ 19 + 5 + 2 ] / t =
26 / t
t = 1 , 2 , 13, 26
(5N) / ( N + 7) = 2
5N = 2 (N + 7)
5N = 2N + 14
3N = 14
N = 14/3
\( (\sqrt{2}+\sqrt{3}+1)^3\)
\(a\sqrt{2} + b\sqrt{3} + c\sqrt{6} + d \)
[ sqrt 2 + sqrt 3 + 1)^3 =
16 + 14 sqrt(2) + 12 sqrt(3) + 6 sqrt(6)
a= 14
b = 12
c = 6
d = 16
Sum = 48
y = mx + b
m = slope = (5- -11) /( -4-1) = -16/5
Using the slope and (1,-11) we have
y = (-16/5) (x -1) -11
y = (-16/5)x + 16/5 - 11
y = (-16/5)x - 39/5
(16/5)x + y = -39/5 ...... multiply through by (-5/39)
(-16/39)x - (5/39)y = 1
And we can write
x / ( -39/16) + y / ( -39/5) = 1
a = -39/16
