What is the fourth term in the geometric sequence 3, –9, 27, … ?

The volume of the cylinder is approximately 40,107.6 cubic yards.

What is the volume of the cylinder?

The formula for the volume of a right circular cylinder is:

[tex]V = \pi r^2h[/tex]

The formula for the volume of a right circular cylinder is:

[tex]V = \pi r^2h[/tex]

Substituting the given values:

V = 3.14 x 17.2² x 45.3

V = 3.14 x 296.84 x 45.3

V = 40,107.6152 cubic yards

Rounding to the nearest tenth:

V ≈ 40,107.6 cubic yards

Therefore, the volume of the cylinder is approximately 40,107.6 cubic yards.

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Answer: 42080.87328 or 42,080.9 rounded to the nearest tenth

Step-by-step explanation:

V=πr2

V= 3.14 x 17.2 x 45.3

V= 3.14 x 17.2 squared x 45.3

= 17.2 squared is 295.84

V= 3.14 x 295.84 x 45.3

V= 42,080.87328

round it to nearest tenth and get 42,080.9 yd

This |x - 32| <= 2 means that the tire pressure can be anywhere between 30 psi and 34 psi.

In order for Ms. Sartain's car to have optimal gas mileage, the tire pressure should remain within 2 psi of 32 psi at all times. This can be modeled with an absolute value inequality.

The absolute value inequality that models this situation is |x - 32| <= 2. This inequality states that the difference between the tire pressure, x, and the optimal pressure, 32, should be less than or equal to 2.

In other words, the tire pressure can be 2 psi above or below the optimal pressure of 32 psi and still be within the acceptable range. This means that the tire pressure can be anywhere between 30 psi and 34 psi.

So the correct answer is |x - 32| <= 2.

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a)P(number greater than 8) = 4/12 = 1/3 ≈ 0.3

b)P(number less than 6) = 5/12 ≈ 0.4

c)If each face has an equal chance of rolling, the solid is fair. The solid is fair since the faces are numbered sequentially from 1 to 12 and each face has an equal chance of being rolled.

One of the number types in algebra that has a whole integer and a fractional portion separated by a decimal point is a decimal. The decimal point is the dot that appears between the parts of a whole number and a fraction. An example of a decimal number is 34.5.

from the question:

a) A solid has 12 equal-sized faces with numbers ranging from 1 to 12. The chance of getting a number larger than 8 is calculated by dividing the total number of faces by the number of faces with numbers greater than 8. Given that there are 4 faces (12 - 8) with numbers greater than 8, the likelihood of drawing one is:

P(number more than 8) = 4/12 = 1/3 =  0.35

b) Similarly, the chance of receiving a number less than 6 is calculated by dividing the total number of faces by the number of faces that have numbers less than 6. Given that there are 6 - 1 = 5 faces with numbers lower than 6, the likelihood of drawing one is as follows:

P(less than six) = 5/12=  0.4

c) If each face has an equal chance of rolling, the solid is fair. The solid is fair since the faces are numbered sequentially from 1 to 12 and each face has an equal chance of being rolled.

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The exponential function showing the relationship between y and t is y = 16,000(0.78)^t

From the question, we have the following parameters that can be used in our computation:

Initial value, a = 16000

Rate = 22% decrement

The exponential function for the resale value y of the car, in dollars, after t years since the purchase can be expressed as:

y = a(1 - r)^t

Substitute the known values in the above equation, so, we have the following representation

y = $16,000 x (1 - 0.22)^t

Evaluate

y = 16,000(0.78)^t

Where 0.78 is the factor by which the resale value decreases each year, calculated as (100% - 22%) / 100% = 0.78.

Hence, the function is y = 16,000(0.78)^t

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The product of these two matrices is a 3x3 matrix, AB.
AB = [[-9, -7, -1]
     [-9, -42, -7]
     [63, -42, 7]]

To determine AB, we need to multiply matrix A and matrix B. The dimensions of matrix A are 3x1 and the dimensions of matrix B are 1x3. Since the number of columns in matrix A is equal to the number of rows in matrix B, we can multiply these matrices. The resulting matrix will have the dimensions of the number of rows in matrix A and the number of columns in matrix B, which is 3x3.

To multiply the matrices, we take the dot product of each row in matrix A with each column in matrix B. The dot product is the sum of the products of the corresponding entries in the row and column.

AB = [[(-1)(-9) + (-6)(-7) + (7)(-1)], [(-1)(-9) + (-6)(-7) + (7)(-1)], [(-1)(-9) + (-6)(-7) + (7)(-1)]]
AB = [[-9, -7, -1]
     [-9, -42, -7]
     [63, -42, 7]]

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The probability that the selected chocolate will be either cookies and cream or peanut butter cups are,

let cookies and cream be x

and peanut butter cups be y

As these are the two chocolates in the bag,

there is a 50:50 probability

Hence,

The probability of cookies and cream = 50%

The probability of peanut butter cups=50%

As x+y=total both have equal probability

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Based on the probability distribution, the probability of a randomly selected account having 4 devices is 0.08. The probability of a randomly selected account having at least 3 devices is 0.44. The probability of a randomly selected account having 2 or 4 devices is 0.51. The mean number of devices per account is 2.39. The standard deviation of the distribution is 1.108. The probability that the number of devices in a randomly selected account lies within one standard deviation of the mean (inclusive) is 0.80.

For the given probability distribution, the probability of a randomly selected account having 4 devices is 0.08. This is because the total probability of all possible outcomes must equal 1. So, we can find the missing probability by subtracting the probabilities of the other outcomes from 1:

1 - 0.13 - 0.43 - 0.29 - 0.07 = 0.08

The probability of a randomly selected account having at least 3 devices is the sum of the probabilities of having 3, 4, or 5 devices:

0.29 + 0.08 + 0.07 = 0.44

The probability of a randomly selected account having 2 or 4 devices is the sum of the probabilities of having 2 and 4 devices:

0.43 + 0.08 = 0.51

The mean number of devices per account can be found by multiplying each possible outcome by its probability and summing the results:

(1)(0.13) + (2)(0.43) + (3)(0.29) + (4)(0.08) + (5)(0.07) = 2.39

The standard deviation of the distribution can be found by first calculating the variance and then taking the square root:

Variance = (1-2.39)^2(0.13) + (2-2.39)^2(0.43) + (3-2.39)^2(0.29) + (4-2.39)^2(0.08) + (5-2.39)^2(0.07) = 1.2279

Standard deviation = √1.2279 = 1.108

The probability that the number of devices in a randomly selected account lies within one standard deviation of the mean (inclusive) is the sum of the probabilities of the outcomes that fall within this range:

0.43 + 0.29 + 0.08 = 0.80

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a. The general exponential growth equation is given by: y = abˣ

b. The general exponential decay equation is given by: [tex]y = a (1 - r)^x[/tex]

Exponential growth is a type of growth pattern in which a quantity grows at an increasing rate proportional to its current value. This means that the larger the quantity, the faster it grows.

a. The general exponential growth equation is given by:

 y = abˣ

Where y is the final value, 'a' is the initial value, b is the growth factor or base, and x is the time or number of periods.

Exponential decay is a type of decay pattern in which a quantity decreases at a decreasing rate proportional to its current value. This means that the larger the quantity, the slower it decays.

b. The general exponential decay equation is given by:

[tex]y = a (1 - r)^x[/tex]

Where y is the final value, a is the initial value, r is the decay rate, and x is the time or number of periods

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For given functions, "The y-intercept of f(x) is equal to the y-intercept of g(x)" is the correct answer i.e. A.

What is the definition of a function?

In mathematics, a function is a relation between two sets of elements, called the domain and the range, such that each element in the domain corresponds to exactly one element in the range.

More specifically, a function is a rule that assigns each element of the domain (input) to a unique element in the range (output). The notation for a function f with domain D and range R is typically written as:

f: D → R

f is a function mapping elements from the domain D to elements in the range R.

Now,

To find the y-intercept of a function, we set x = 0 and evaluate the function.

For [tex]f(x) = -4(1.09)^x[/tex], when x = 0, we get:

[tex]f(0) = - 4(1.09)^0 = - 4[/tex]

So, the y-intercept of f(x) is -4.

For g(x), we are given a table of values, and we can see that when x = 0, g(x) = -4. Therefore, the y-intercept of g(x) is also -4.

Hence,

          The first option "The y-intercept of f(x) is equal to the y-intercept of g(x)" is the correct answer.

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The remainder when -233 is divided by 11 is 9. To find the remainder when a is divided by b, we can use the formula:

r = a % b

Where % is the modulo operator, which gives the remainder when one number is divided by another.

In this case, we have a = -233 and b = 11. Plugging these values into the formula, we get:

r = -233 % 11

Using a calculator or doing the division by hand, we find that the remainder is -2. However, since we are looking for the positive remainder, we can add b to this value to get the correct answer:

r = -2 + 11 = 9

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Given that the polynomial has a zero at 5 with multiplicity 2, its complete factorization is (x - 5)²(z + 9)(z - 2).

To factor the given polynomial z⁴ - 3z³ - 63z² + 355z - 450, given that it has a zero at 5 with multiplicity 2, we can use the fact that (z - 5)² is a factor of the polynomial. We can then use synthetic division to find the other factors.

First, we divide the polynomial by (z - 5) using synthetic division:
5   |   1  -3  -63  355  -450
    |       5    10  -265  450
        1   2  -53    90       0

The result of the division is z³ + 2z² - 53z + 90 . Divide it by (z - 5) again since the multiplicity is 2.
5   |   1   2  -53    90
    |       5    35   -90

        1   7   -18      0

The result of the second division is z² + 7z - 18. This polynomial can still be factorized as (z + 9)(z - 2).

So, the final answer is:

z⁴ - 3z³ - 63z² + 355z - 450 = (x - 5)²(z + 9)(z - 2).

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The measure of <TAV is [tex]92^{o}[/tex].

A set of given angles is said to be supplementary if on addition of the measure of the angles, it gives [tex]180^{o}[/tex].

In the given diagram, given that PQ is parallel to RS, it can be observed that;

<SCU ≅ <ACB (definition of vertically opposite angles)

Thus, <ACB = [tex]18^{o}[/tex]

<QAC = <BCA     (alternate angle property)

So that,

<QAC = [tex]18^{o}[/tex]

But,

<TAP ≅ <QAC  (definition of vertically opposite angles)

So that;

<BAP + <PAT + <TAV = [tex]180^{o}[/tex]  (sum of angles on a straight line)

Then,

70 + 18 + <TAV = 180

<TAV = 180 - 88

         = 92

<TAV = [tex]92^{o}[/tex]

The measure of <TAV = [tex]92^{o}[/tex].

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Answer:

1.5x + 2y = 12

Step-by-step explanation:

The equation representing Serena’s spending on snacks today would be 1.5x + 2y = 12, where x represents the number of drinks purchased and y represents the number of bags of chips purchased.

Therefore, the equation is 1.5x + 2y = 12.

The basis for span((1,−1,2,2),(2,2,1,1),(2,−1,−1,0),(4,2,−5,−3)) is {(1,−1,2,2), (2,2,1,1), (2,−1,−1,0), (4,2,−5,−3)}.

A basis for a vector space is a set of linearly independent vectors that span the vector space. In this case, we need to find a basis for the vector space spanned by the given vectors (1,−1,2,2), (2,2,1,1), (2,−1,−1,0), and (4,2,−5,−3).

To find a basis, we can use the row reduction method. First, we write the given vectors as rows of a matrix:

```
1 -1  2  2
2  2  1  1
2 -1 -1  0
4  2 -5 -3
```

Next, we use row operations to reduce the matrix to row echelon form:

```
1 -1  2  2
0  4 -3 -3
0  0 -5 -4
0  0  0  2
```

Now, we can see that the first, second, third, and fourth rows are all linearly independent (since they all have a leading 1 in a different column). Therefore, the original vectors (1,−1,2,2), (2,2,1,1), (2,−1,−1,0), and (4,2,−5,−3) form a basis for the vector space.

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Answers:

7. x=4.8          8. x=36.6       9. x=36.9  10. x=17.8

Work is shown in the picture below, I'm learning this stuff right now too, so I hope it helps!

The correct answer is f^-1(x) = (8-√x)/2; f^-1 is a function.

To find the inverse of the function f(x) = (8-2x)^2, we need to switch the x and y variables and solve for y. This will give us f^-1(x).

So, we start with:

x = (8-2y)^2

Next, we take the square root of both sides:

√x = 8-2y

Then, we isolate the y variable:

2y = 8-√x

y = (8-√x)/2

So, the inverse of the function is:

f^-1(x) = (8-√x)/2

Now, we need to determine whether f^-1(x) is a function. To do this, we can use the horizontal line test. If a horizontal line intersects the graph of f^-1(x) at more than one point, then f^-1(x) is not a function.

In this case, a horizontal line will only intersect the graph of f^-1(x) at one point, so f^-1(x) is a function.

Therefore, the correct answer is f^-1(x) = (8-√x)/2; f^-1 is a function.

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2. The given quadratic equation is in the general form:

ax² + bx + c = 0

therefore:

a = 2

b = -4

c = -3

The quadratic formula is thus:

[tex]x=\frac{-b(+-)\sqrt{b^2-4ac} }{2a}[/tex]

Substituting the values found for a, b, and c:

[tex]x=\frac{-(-4)+\sqrt{(-4)^2-4(2)(-3)} }{2(2)}[/tex] and [tex]x=\frac{-(-4)-\sqrt{(-4)^2-4(2)(-3)} }{2(2)}[/tex]

Therefore x = 2.58, x = -0.58

3. Using the same method as above, first, bring all values to one side, leaving the RHS = 0

a = 1

b = 2

c = -1

The quadratic formula is thus:

[tex]x=\frac{-b(+-)\sqrt{b^2-4ac} }{2a}[/tex]

Substituting the values found for a, b, and c:

[tex]x=\frac{-(2)+\sqrt{(2)^2-4(1)(-1)} }{2(1)}[/tex] and [tex]x=\frac{-(2)-\sqrt{(2)^2-4(1)(-1)} }{2(1)}[/tex]

Therefore, x = 0.41, x = -2.41

[tex]2 {x}^{2} - 4x - 3 = 0[/tex]

A Here ,

[tex]\boxed{a = 2 }\\\boxed{b = - 4} \\ \boxed{c = - 3}[/tex]

B Filling in the values of a , b and c in the Quadratic formula below , we get

[tex]x = \frac{- (b)\pm \sqrt{( {b}^{2}) - 4(a)(c) } }{2(a)} \\ [/tex]

C Simplifying each section , we get

[tex]x = \frac{ - ( - 4) + \sqrt{( { - 4}^{2} ) - 4(2)( - 3)} }{2 \times 2} [/tex]

or

[tex]x = \frac{ - ( - 4) - \sqrt{ {( - 4})^{2} - 4(2)( - 3) } }{2 \times 2} [/tex]

D Simplifying answers from Part C , we get

[tex]\boxed{x = \frac{2 + \sqrt{10} }{2}} \: \: \: \: or \: \: \: \: \boxed{ x = \frac{2 - \sqrt{10} }{2} } \\ [/tex]

Therefore ,

[tex]\boxed{x = 2.58} \: \: \: \: and \: \: \: \: \boxed{x = - 0.58}[/tex]

_____________________________________

[tex] {x}^{2} + 2x = 1 \\ \implies \: {x}^{2} + 2x - 1 = 0[/tex]

A Here ,

[tex]\boxed{a = 1} \\ \boxed{b = 2} \\ \boxed{c = - 1}[/tex]

B Filling in the values of a , b and c in the Quadratic formula below , we get

[tex]x = \frac{- (b)\pm \sqrt{( {b}^{2}) - 4(a)(c) } }{2(a)} \\ [/tex]

C Simplifying each section , we get

[tex]x = \frac{ - (2) + \sqrt{ ({2}^{2} ) - 4(1)( - 1)} }{2 \times 1} [/tex]

or

[tex]x = \frac{ - (2) - \sqrt{( {2}^{2}) - 4(1)( - 1) } }{2 \times 1} [/tex]

D Simplifying answers from Part C , we get

[tex]\boxed{x = - 1 + \sqrt{2} } \: \: \: \: or \: \: \: \: \boxed{x = - 1 - \sqrt{2} }[/tex]

Therefore

[tex]\boxed{x = 0.41} \: \: \: \: or \: \: \: \: \boxed{x = -2.41 }[/tex]

hope helpful! :)

[tex]x=\textit{kgs of solution at 30\%}\\\\ ~~~~~~ 30\%~of~x\implies \cfrac{30}{100}(x)\implies 0.3 (x) \\\\\\ y=\textit{kgs of solution at 15\%}\\\\ ~~~~~~ 15\%~of~y\implies \cfrac{15}{100}(y)\implies 0.15 (y) \\\\\\ \textit{60 kgs of solution at 20\%}\\\\ ~~~~~~ 20\%~of~60\implies \cfrac{20}{100}(60)\implies 0.2 (60)\implies 12 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{array}{lcccl} &\stackrel{kgs}{quantity}&\stackrel{\textit{\% of kgs that is}}{\textit{nitrogen only}}&\stackrel{\textit{kgs of}}{\textit{nitrogen only}}\\ \cline{2-4}&\\ \textit{1st Fert.}&x&0.3&0.3x\\ \textit{2nd Fert.}&y&0.15&0.15y\\ \cline{2-4}&\\ mixture&60&0.2&12 \end{array}~\hfill \begin{cases} x + y = 60\\\\ 0.3x+0.15y=12 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{using the 1st equation}}{x+y=60}\implies y=60-x \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{using the 2nd equation}}{0.3x+0.15y=12}\implies \stackrel{\textit{substituting from above}}{0.3x+0.15(60-x)=12} \\\\\\ 0.3x+9-0.15x=12\implies 0.15x=3\implies x=\cfrac{3}{0.15} \\\\\\ \boxed{x=20}\hspace{5em}\stackrel{ 60~~ - ~~20 }{\boxed{y=40}}[/tex]

The graph is symmetric about the y-axis, so its a even function.

EVEN function:

ODD function:

Thus, the graph is symmetric about the y-axis, so its a even function.

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a) The exact distance is 5√2.

b) The midpoint of the line segment is (5.5, 1.5).

Part 1 of 2 (a) The exact distance between the points (9,2) and (2,1) can be found using the distance formula:

Distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Plugging in the given values:

Distance = √[(2 - 9)^2 + (1 - 2)^2]

Simplifying:

Distance = √[(-7)^2 + (-1)^2]

Distance = √[49 + 1]

Distance = √50

Distance = 5√2

Therefore, the exact distance between the points is 5√2.

Part 2 of 2 (b) The midpoint of the line segment whose endpoints are the given points can be found using the midpoint formula:

Midpoint = [(x1 + x2)/2, (y1 + y2)/2]

Plugging in the given values:

Midpoint = [(9 + 2)/2, (2 + 1)/2]

Simplifying:

Midpoint = [11/2, 3/2]

Midpoint = (5.5, 1.5)

Therefore, the midpoint is (5.5, 1.5).

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The error that occurred is called a transposition error.

A transposition error is when two digits are reversed or transposed in an accounting transaction. In this case, the debit that was supposed to be made to the Purchaser account was instead debited to the Accounts Payable account.

To correct this error, we need to make a journal entry that reverses the incorrect entry and then make the correct entry. The journal entry to reverse the incorrect entry would be:

Debit: Accounts Payable $5,000
Credit: Purchaser $5,000

This entry reverses the incorrect debit to Accounts Payable and the incorrect credit to Purchaser.

Next, we need to make the correct entry, which is:

Debit: Purchaser $5,000
Credit: Accounts Payable $5,000

This entry correctly debits the Purchaser account and credits the Accounts Payable account.

After these two journal entries are made, the accounts will be correctly balanced and the error will be corrected.

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complete question

A P^(5),000 debit to be made to the Purchaser account was debited to Accounts payabhe instead. which type of error is found here?

The factored form of the given quadratic equation is (2x + 7)(x - 2).

To factor a quadratic equation with a leading coefficient, we need to find two numbers that multiply to give us the constant term (-14) and add to give us the middle term (3).

In this case, the two numbers are 7 and -2. We can then use these numbers to rewrite the middle term of the equation and then factor by grouping.

Here are the steps to factor the given quadratic equation:

1. Rewrite the equation with the new middle terms: 2x^(2) + 7x - 2x - 14
2. Group the first two terms and the last two terms: (2x^(2) + 7x) + (-2x - 14)
3. Factor out the greatest common factor from each group: x(2x + 7) - 2(2x + 7)
4. Factor out the common binomial: (2x + 7)(x - 2)

So, the factored form of the given quadratic equation is (2x + 7)(x - 2).

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Answer:

The number is → 56

Step-by-step explanation:

tens digit [tex]\Rightarrow x[/tex]

unit digit [tex]\Rightarrow y[/tex]

"The difference between the digits of a two-digit number is 1...", " ...the unit digit is greater than the tens digit..."

[tex]y-x=1 \qquad \textbf{ec.1}[/tex]

"The number itself is one more (unit) than five times the sum of its digits..."

[tex]10x+y=5(x+y)+1\\ 10x+y= 5x + 5y+1\\5x= 4y+1 \qquad \textbf{ec.2}[/tex]

we clear "y" in equation 1:

[tex]y=1+x \qquad \textbf{ec.3}[/tex]

then we substitute in equation 2:

[tex]5x=4(1+x)+1\\5x=5+4x\\\boxed{x=5}[/tex]

Finally, we substitute in equation 3:

[tex]y=1+5\\\boxed{y=6}[/tex]

With this we have solved the exercise.

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

Answer:

they finished the entire pie

Step-by-step explanation:

The standard matrix for the stated composition of linear operators on R2 is:

The standard matrix for the stated composition of linear operators on R2 can be found by multiplying the matrices for each individual operation.

First, let's find the matrix for a rotation of 270° counterclockwise:

Answer:

for the first, the answers are 1/2, 1, 2, 4, and 8. for the second, 22[tex]\frac{1}{2}[/tex] sq. km.

Step-by-step explanation:

1/4 times 2 is 1/2, times 2 is 1, times 2 is 2, times two is 4, time 2 is 8.

for the second one, area = base times height. 6 3/4 times 3 1/3 is 22 1/2 km squared.

Answer: 5,948 feet

Step-by-step explanation:

All you really need to do in this problem is subtract 22,834 - 16,896 as shown below:

22,834 - 16,836 = 5,948 feet

Answer:

Step-by-step explanation:

You want the weights of trains A and B if the sum of their weights is 184 tons and the difference of their weights is 90 tons.

We can write the equations for the weights as ...

  A +B = 184

  A -B = 90

Adding the two equations gives ...

  2A = 274

  A = 137

Subtracting the second equation from the first gives ...

  2B = 94

  B = 47

Train A weighs 137 tons; train B weighs 47 tons.

<95141404393>

end{bmatrix} = \begin{bmatrix} 1 \\ \frac{1}{5} \\ 1 \end{bmatrix}.

(a) Let $a^1 = \begin{bmatrix} 1 \\ 1 \\ 2 \\ 1 \end{bmatrix}, a^2 = \begin{bmatrix} -1 \\ 2 \\ 0 \\ -2 \end{bmatrix},$ and $a^3 = \begin{bmatrix} 1 \\ 4 \\ 4 \\ 0 \end{bmatrix}.$ Write the matrix $A = \begin{bmatrix} a^1 & a^2 & a^3 \end{bmatrix}$ in the form $A = QR$ by using the Gram-Schmidt process. (b) Use the QR factorization of $A$ in part (a) to solve the equation $Ax = b,$ where $b = \begin{bmatrix} 3 \\ 1 \\ 2 \\ 1 \end{bmatrix}.$The Gram-Schmidt algorithm is a numerical method to produce orthonormal basis of a subspace in Hilbert space that spans the same space, which makes the basis more convenient to work with. As for the first part of the question, let us begin by applying the Gram-Schmidt algorithm to $a^1, a^2, a^3.$ We begin by defining $q_1 = a^1 / \|a^1\|.$ Hence,$$q_1 = \frac{1}{3}\begin{bmatrix} 1 \\ 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 1/3 \\ 1/3 \\ 2/3 \\ 1/3 \end{bmatrix}.$$Next, we define $v_2 = a^2 - \langle q_1, a^2 \rangle q_1.$ Therefore,$$v_2 = a^2 - \frac{-1}{3}(1/3)q_1 = \begin{bmatrix} -7/9 \\ 8/9 \\ -2/9 \\ -4/9 \end{bmatrix}.$$Now, we can define $q_2 = v_2 / \|v_2\|.$ Thus,$$q_2 = \frac{1}{3}\begin{bmatrix} -7 \\ 8 \\ -2 \\ -4 \end{bmatrix}.$$Finally, we define $v_3 = a^3 - \langle q_1, a^3 \rangle q_1 - \langle q_2, a^3 \rangle q_2.$ Then,$$v_3 = a^3 - \frac{5}{9}q_1 - \frac{7}{27}q_2 = \begin{bmatrix} -1/27 \\ 5/9 \\ 22/27 \\ -5/27 \end{bmatrix}.$$Lastly, we can define $q_3 = v_3 / \|v_3\|,$ so$$q_3 = \frac{1}{3}\begin{bmatrix} -1 \\ 5 \\ 22 \\ -5 \end{bmatrix}.$$Now, we can write $A = QR$ as $$\begin{bmatrix} a^1 & a^2 & a^3 \end{bmatrix} = \begin{bmatrix} q_1 & q_2 & q_3 \end{bmatrix} \begin{bmatrix} r_{11} & r_{12} & r_{13} \\ 0 & r_{22} & r_{23} \\ 0 & 0 & r_{33} \end{bmatrix}.$$We can obtain the entries of the $R$ matrix by calculating the inner product of each $q_i$ with $a^j.$ Thus,$$r_{11} = \|a^1\| = \sqrt{7},$$$$r_{12} = \langle q_1, a^2 \rangle = \frac{-1}{3}\sqrt{7},$$$$r_{13} = \langle q_1, a^3 \rangle = \frac{5}{9}\sqrt{7},$$$$r_{22} = \|v_2\| = \frac{5}{3}\sqrt{2},$$$$r_{23} = \langle q_2, a^3 \rangle = \frac{-7}{9}\sqrt{2},$$$$r_{33} = \|v_3\| = \frac{2}{3}\sqrt{6}.$$Therefore,$$\begin{bmatrix} a^1 & a^2 & a^3 \end{bmatrix} = \begin{bmatrix} q_1 & q_2 & q_3 \end{bmatrix} \begin{bmatrix} \sqrt{7} & -\frac{1}{3}\sqrt{7} & \frac{5}{9}\sqrt{7} \\ 0 & \frac{5}{3}\sqrt{2} & -\frac{7}{9}\sqrt{2} \\ 0 & 0 & \frac{2}{3}\sqrt{6} \end{bmatrix}.$$Now, let us solve the equation $Ax = b$ by using the QR factorization of $A.$ We can write $Ax = QRx = b.$ Since $Q$ is orthogonal, we can multiply both sides of the equation by $Q^T$ to obtain $Rx = Q^Tb.$ Note that $Q^Tb$ is easy to compute since $Q^T$ is just the matrix with the $q_i$'s as rows. Thus,$$\begin{bmatrix} \sqrt{7} & -\frac{1}{3}\sqrt{7} & \frac{5}{9}\sqrt{7} \\ 0 & \frac{5}{3}\sqrt{2} & -\frac{7}{9}\sqrt{2} \\ 0 & 0 & \frac{2}{3}\sqrt{6} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} \frac{2}{3} \\ \frac{1}{3} \\ \frac{2}{3} \end{bmatrix}.$$This gives the system of equations$$\begin{cases} \sqrt{7}x_1 - \frac{1}{3}\sqrt{7}x_2 + \frac{5}{9}\sqrt{7}x_3 = \frac{2}{3}, \\ \frac{5}{3}\sqrt{2}x_2 - \frac{7}{9}\sqrt{2}x_3 = \frac{1}{3}, \\ \frac{2}{3}\sqrt{6}x_3 = \frac{2}{3}. \end{cases}$$Solving the last equation for $x_3,$ we obtain $x_3 = 1.$ Substituting this into the second equation, we obtain $x_2 = \frac{1}{5}.$ Finally, substituting these values into the first equation gives us $x_1 = 1.$ Therefore,$$x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 1 \\ \frac{1}{5} \\ 1 \end{bmatrix}.$$

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