Please help me out :((

1) 5*x + 6 = 2 + 3*x

5*x - 3*x = 2 - 6

(5 - 3)*x = -4

2*x = - 4

x = -4/2 = -2

x = -2

We have only one solution.

2) 2*(6 - 2*y) = -1*(4*y - 9)

12 - 4*y = -4*y + 9

12 - 9 = -4*y + 4*y = 0

3 = 0

This is absurd, so this equation has no solution.

3) 2*z - 6 = 2*(z + 2) - 10

2*z - 6 = 2*z + 4 - 10

2*z - 6 = 2*z - 6

Here we have the exact same thing on both sides of the equation, then we have infinite solutions, this happens because z can take any value, and the equation will be true always.

4) We want to have no solutions, so we need to end with something like:

1 = 6

so, we start with:

5*x + 1 = 5*x + A

We want to find the value of A.

First, we can subtract 5*x in both sides, so we get:

1 = A

Now we just need to take A different than 1, for example, if A  = 2, then:

1 = 2

In this case the equation has no solutions, then we have:

5*x + 1 = 5*x + 2

5) We want to have one solution:

3*x - 3 = A*x + 11

We want to end with something like x = k

Let's solve the equation for x:

3*x - 3 = A*x + 11

3*x - A*x = 11 + 3

(3 - A)*x = 14

if we take A = 2, then:

(3 - 2)*x = 14

x = 14

So this equation has only one solution, the equation is:

3*x - 3 = 2*x + 11

6) We want to have infinitely many solutions, then we need to have the same thing on both sides, then we need to have an equation like:

8*x - 7 = 8*x - 7

step-by-step explanation:

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).