I:2x – y + z = 7
II:x + 2y – 5z = -1
III:x – y = 6
you can first use III and substitute x or y to eliminate it in I and II (in this case x):
III: x=6+y
-> substitute x in I and II:
I': 2*(6+y)-y+z=7
12+2y-y+z=7
y+z=-5
II':(6+y)+2y-5z=-1
3y+6-5z=-1
3y-5z=-7
then you can subtract II' from 3*I' to eliminate y:
3*I'=3y+3z=-15
3*I'-II':
3y+3z-(3y-5z)=-15-(-7)
8z=-8
z=-1
insert z in II' to calculate y:
3y-5z=-7
3y+5=-7
3y=-12
y=-4
insert y into III to calculate x:
x-(-4)=6
x+4=6
x=2
so the solution is
x=2
y=-4
z=-1
Answer:
A) 952 insects are in the original colony
B) 10494 insects are in the colony after 20 weeks
C) 25 weeks
Explanations:
The model representing the population of the species of insects
![P(t)=952e^{0.12t}](https://tex.z-dn.net/?f=P%28t%29%3D952e%5E%7B0.12t%7D)
An exponential growth is of the form:
![P(t)=P_0e^{kt}](https://tex.z-dn.net/?f=P%28t%29%3DP_0e%5E%7Bkt%7D)
where P₀ is the original population
t is the time taken in weeks
Comparing the two equations:
![P_0=\text{ 952}](https://tex.z-dn.net/?f=P_0%3D%5Ctext%7B%20952%7D)
952 insects are in the original colony
B. The number of insects that will be in the colony after 20 weeks
Substituting t = 20 into the function given
![\begin{gathered} P=952e^{0.12(20)} \\ \text{P = }10494 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%3D952e%5E%7B0.12%2820%29%7D%20%5C%5C%20%5Ctext%7BP%20%3D%20%7D10494%20%5Cend%7Bgathered%7D)
C) If the population, P = 20000
2x+y=20 and -5y=6x+12 are your equations.
You can turn 2x+y=20 into y=-2x+20
Substitute this value of y into the other equation.
-5(-2x+20)=-6x+12
10x-100=-6x+12
16x=112
x=7
Substitute this value of x into the other equation.
2(7)+y=20
y=6
Your answer is x=7 and y=6.
Times it and find the solution to the problem to divide all numbers