Answer:
-0.4
Step-by-step explanation:
y=-0.4x+7
Answer:
a) -4
b) 1
c) 1
Step-by-step explanation:
a) The matrix A is given by:
![A=\left[\begin{array}{ccc}-3&0&1\\2&-4&2\\-3&-2&1\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-3%260%261%5C%5C2%26-4%262%5C%5C-3%26-2%261%5Cend%7Barray%7D%5Cright%5D)
to find the eigenvalues of the matrix you use the following:
where lambda are the eigenvalues and I is the identity matrix. By replacing you obtain:
![A-\lambda I=\left[\begin{array}{ccc}-3-\lambda&0&1\\2&-4-\lambda&2\\-3&-2&1-\lambda\end{array}\right]](https://tex.z-dn.net/?f=A-%5Clambda%20I%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-3-%5Clambda%260%261%5C%5C2%26-4-%5Clambda%262%5C%5C-3%26-2%261-%5Clambda%5Cend%7Barray%7D%5Cright%5D)
and by taking the determinant:
![[(-3-\lambda)(-4-\lambda)(1-\lambda)+(0)(2)(-3)+(2)(-2)(1)]-[(1)(-4-\lambda)(-3)+(0)(2)(1-\lambda)+(2)(-2)(-3-\lambda)]=0\\\\-\lambda^3-6\lambda^2-12\lambda-16=0](https://tex.z-dn.net/?f=%5B%28-3-%5Clambda%29%28-4-%5Clambda%29%281-%5Clambda%29%2B%280%29%282%29%28-3%29%2B%282%29%28-2%29%281%29%5D-%5B%281%29%28-4-%5Clambda%29%28-3%29%2B%280%29%282%29%281-%5Clambda%29%2B%282%29%28-2%29%28-3-%5Clambda%29%5D%3D0%5C%5C%5C%5C-%5Clambda%5E3-6%5Clambda%5E2-12%5Clambda-16%3D0)
and the roots of this polynomial is:
hence, the real eigenvalue of the matrix A is -4.
b) The multiplicity of the eigenvalue is 1.
c) The dimension of the eigenspace is 1 (because the multiplicity determines the dimension of the eigenspace)
Answer:
1,439.5 miles
Step-by-step explanation:
The odometer is the number of miles on the car throughout its entire lifespan. When a car is new and just built, the odometer has a reading of 0. As you drive, each mile is added on the odometer. If you start your trip at 18,594.3 and you travel and stop when it reads 20,023.8, then the difference between the two is you total miles traveled.
20,023.8
-18,594.3
________
1,439.5
You traveled 1,439.5 miles.
Let x be rate of boat in still water
let y be rate of current
we use this equation to relate quantities:
distance = speed · time
we have two unknowns so we might need to create a system of equationss
upstream:
speed (in km/h) = x - y
(we get speed of boat then subtract the current's speed from it since current is going against boat direction)
time = 3 hours
distance = 144 km
downstream:
speed (in hm/h) = x + y
(we get speed of boat then add the current's spd from it since current is going against boat direction)
time = 2 hours
distance = 144 km (same distance upstream and downstream)
using distance = speed times time
for upstream
144 = 3(x-y)
144 = 3x - 3y
for downstream
144 = 2(x+y)
72 = x + y
system of eqns:
144 = 3x - 3y
72 = x + y
solve by substitution: move 72 = x + y into x = 72 - y and subst into other equation for x
144 = 3(72 - y) - 3y
144 = 216 - 3y - 3y
144 = 216 - 6y
144 - 216 = -6y
-72 = -6y
y = 12 km/h
Use x = 72 - y to find x with y = 12: x = 72 - 12 = 60 km/h
rate of boat in still water is 60 km/h
rate of the current is 12 km/h
