What is the solution to the system of equations below? y = negative one-third x + 6 and x = –6 (–6, 8) (–6, 4) (8, –6) (4, –6)

Suppose, number of sunshine days in Boston, Massachusetts is: x

Then, number of sunshine days in Colorado Spring, Colorado will be: 1.4 x

Given that, total number of days of sunshine for both city is: 475

That is, x+1.4x = 475

Solving for x,

x=198

1.4x= 277

Therefore,

Number of sunshine days in Boston, Massachusetts is 198

and

Number of sunshine days in Colorado Spring, Colorado is 277

Answer:

# The solution x = -5

# The solution is x = 1

# The solution is x = 6.4

# The solution is x = 4

# The solution is 1.7427

# The solution is 0.190757

Step-by-step explanation:

* Lets revise some rules of the exponents and the logarithmic equation

# Exponent rules:

1- b^m  ×  b^n  =  b^(m + n) ⇒ in multiplication if they have same base

  we add  the power

2- b^m  ÷  b^n =  b^(m – n) ⇒  in division if they have same base we

   subtract  the power

3- (b^m)^n = b^(mn) ⇒ if we have power over power we multiply

   them

4- a^m × b^m = (ab)^m ⇒ if we multiply different bases with same  

   power then we multiply them ad put over the answer the power

5- b^(-m) = 1/(b^m)  (for all nonzero real numbers b) ⇒ If we have

   negative power we reciprocal the base to get positive power

6- If  a^m  =  a^n  ,  then  m  =  n ⇒ equal bases get equal powers

7- If  a^m  =  b^m  ,  then  a  =  b    or    m  =  0

# Logarithmic rules:

1-

2-

3-

4-

5-

* Now lets solve the problems

#

- Change the base 9 to 3²

∴

∴

- Same bases have equal powers

∴ x + 1 = 2x + 6 ⇒ subtract x and 6 from both sides

∴ 1 - 6 = 2x - x

∴ -5 = x

* The solution x = -5

# ㏒(9x - 2) = ㏒(4x + 3)

- If ㏒(a) = ㏒(b), then a = b

∴ 9x - 2 = 4x + 3 ⇒ subtract 4x from both sides and add 2 to both sides

∴ 5x = 5 ⇒ divide both sides by 5

∴ x = 1

* The solution is x = 1

#

- Use the 1st rule in the logarithmic equation

∴ 6² = 5x + 4

∴ 36 = 5x + 4 ⇒ subtract 4 from both sides

∴ 32 = 5x ⇒ divide both sides by 5

∴ 6.4 = x

* The solution is x = 6.4

#

- Use the rule 3 in the logarithmic equation

∴

- Use the 1st rule in the logarithmic equation

∴ 2² = x(x - 3) ⇒ simplify

∴ 4 = x² - 3x ⇒ subtract 4 from both sides

∴ x² - 3x - 4 = 0 ⇒ factorize it into two brackets

∴ (x - 4)(x + 1) = 0 ⇒ equate each bract by 0

∴ x - 4 = 0 ⇒ add 4 to both sides

∴ x = 4

OR

∵ x + 1 = 0 ⇒ subtract 1 from both sides

∴ x = -1

- We will reject this answer because when we substitute the value

 of x in the given equation we will find and this

 value is undefined, there is no logarithm for negative number

* The solution is x = 4

#

- You can use the calculator directly to find x

∴ x = 1.7427

* The solution is 1.7427

# ⇒ divide the both sides by 2

∴

- Insert ln for both sides

∴

- Use the rule ⇒ ln(e) = 1

∴ 8x = ln(4.6) ⇒ divide both sides by 8

∴ x = ln(4.6)/8 = 0.190757

* The solution is 0.190757

QUESTION 1

This is the same as:

Equate the exponents.

x+1=2(x+3)

Expand:

x+1=2x+6

Group similar terms;

2x-x=1-6

x=-5

QUESTION 2

Equate the arguments.

9x-2=4x+3

Group similar terms;

9x-4x=3+2

5x=5

Divide through by 5

x=1

QUESTION 3

Take antilogarithm to obtain,

This implies that,

5x+4=36

5x=36-4

5x=32

x=32/5

or

QUESTION 4

Use the product rule of logarithms:

Take antilogarithm,

Factor:

This implies that,

But the domain is x>0, therefore the solution is

x=4

QUESTION 5

x=1.7 to the nearest tenth.

QUESTION 6

Divide both sides by 2.

Take natural log of both sides

x=0.2 to the nearest tenth.