Answer: Mark has 28 marbles while Don has 85 marbles.
Step-by-step explanation:
Let x represent the number of marbles that Mark has.
Let y represent the number of marbles that Don has.
If Don has 1 more than 3 times the number of marbles Mark has, it means that
y = 3x + 1
The total number of marbles is 113. It means that
x + y = 113- - - - - - - - - - - - 1
Substituting y = 3x + 1 into equation 1, it becomes
x + 3x + 1 = 113
4x + 1 = 113
4x = 113 - 1 = 112
x = 112/4
x = 28
y = 3x + 1 = 3 × 28 + 1
y = 85
The answer is 0.1 i think
Don’t worry, bud, I gotchu.
Answer: 7.647 but rounded to the second decimal place is 7.65
m = (y2-y1) / (x2-x1)
m= (20000-7000)/(2600-900)
And you know what happens next.
G(x)=-2x-4
f(x)=15+0.75x
the common solution Q is their intersection-> calculate x by setting them equal:
-2x-4=15+0.75x
-2x-(3/4)x=19
-(11/4)x=19
x=-4*19/11
x=-76/11
calculate y:
f(-76/11)=15+(3/4)*(-76/11)
=15+(-3*19/11)
=15+(-57/11)
=15+-5-(2/11)
=10-(2/11)
=108/11
so the solution is a) (-76/11, 108/11)
The given proof of De Moivre's theorem is related to the operations of
complex numbers.
<h3>The Correct Responses;</h3>
- Step C: Expanding and collecting like terms
- Step D: Trigonometric formula for the cosine and sine of the sum of two numbers
<h3>Reasons that make the above selection correct;</h3>
The given proof is presented as follows;
![\mathbf{\left[cos(\theta) + i \cdot sin(\theta) \right]^{k + 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%20%2B%201%7D%7D)
- Step A: By laws of indices, we have;
![\left[cos(\theta) + i \cdot sin(\theta) \right]^{k + 1} = \mathbf{\left[cos(\theta) + i \cdot sin(\theta) \right]^{k} \cdot \left[cos(\theta) + i \cdot sin(\theta) \right]}](https://tex.z-dn.net/?f=%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%20%2B%201%7D%20%3D%20%5Cmathbf%7B%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%7D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%7D)
![\left[cos(\theta) + i \cdot sin(\theta) \right]^{k} \cdot \left[cos(\theta) + i \cdot sin(\theta) \right] = \mathbf{\left[cos(k \cdot \theta) + i \cdot sin(k \cdot \theta) \right] \cdot \left[cos(\theta) + i \cdot sin(\theta) \right]}](https://tex.z-dn.net/?f=%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%7D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%20%3D%20%20%5Cmathbf%7B%5Cleft%5Bcos%28k%20%5Ccdot%20%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Cright%5D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%7D)
- Step B: By expanding, we have;
![\left[cos(k \cdot \theta) + i \cdot sin(k \cdot \theta) \right] \cdot \left[cos(\theta) + i \cdot sin(\theta) \right] = cos(k \cdot \theta) \cdot cos(\theta) - sin(k \cdot \theta) \cdot sin(\theta) + i \cdot \left [sin(k \cdot \theta) \cdot cos(\theta) + cos(k \cdot \theta) \cdot sin(\theta) \right]](https://tex.z-dn.net/?f=%5Cleft%5Bcos%28k%20%5Ccdot%20%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Cright%5D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%20%3D%20cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20-%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%2B%20i%20%20%5Ccdot%20%5Cleft%20%5Bsin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20%2B%20cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D)
- Step D: From trigonometric addition formula, we have;
cos(A + B) = cos(A)·cos(B) - sin(A)·sin(B)
sin(A + B) = sin(A)·cos(B) + sin(B)·cos(A)
Therefore;
![cos(k \cdot \theta) \cdot cos(\theta) - sin(k \cdot \theta) \cdot sin(\theta) + i \cdot \left [sin(k \cdot \theta) \cdot cos(\theta) + cos(k \cdot \theta) \cdot sin(\theta) \right] = \mathbf{ cos(k \cdot \theta + \theta) + i \cdot sin(k \cdot \theta + \theta)}](https://tex.z-dn.net/?f=cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20-%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%2B%20i%20%20%5Ccdot%20%5Cleft%20%5Bsin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20%2B%20cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%20%3D%20%5Cmathbf%7B%20cos%28k%20%5Ccdot%20%5Ctheta%20%2B%20%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28k%20%5Ccdot%20%5Ctheta%20%20%2B%20%5Ctheta%29%7D)
Learn more about complex numbers here:
brainly.com/question/11000934
