Solve the system by substitution, show all steps. -x-y-z=-8 -4+4y+5z=7 2x+2z=4

In Point Form: (x,y,z)= (3,6,-1)

In Equation Form: x=3, y=6, z=-1

Step-by-step explanation:

1) Solve for x

           2x+2z=4 (Subtract 2z from both sides)

           -2z  -2z

           2x= 4-2z  (Divide 2 from both sides)

           /2      /2

           x=2-z

2) Substitute 2-z for x in

                       -x-y-z=-8

                       -(2-z)-y-z=-8

                       -2+z-y-z=-8  (Combine Like Terms)

                       -2-y=-8 (Add 2 to both sides)

                       +2     +2

                       -y=-6  (Divide -1 from both sides)

                       /-1   /-1

                       y=6

3) Substitute 2-z for x & 6 for y in

                          -4x+4y+5z+=7

                          -4(2-z)+4(6)+5z=7  (Get rid of parentheses)

                          -8+4z+24+5z=7   (Combine Like Terms)

                          9z+16=7  (Subtract 16 from both sides)

                               -16  -16

                          9z+=-9  (Divide 9 from both sides)

                          /9     /9

                          z=-1

4) Now find x by substituting -1 for z in

                                          x=2-z

                                          x=2-(-1)  

                                          x=2+1

                                          x=3  

Therefore the answer is:

In Point Form: (x,y,z)= (3,6,-1)

In Equation Form: x=3, y=6, z=-1

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).