Show that y₁(t) = e^ãt cos(μt) and
y₂(t) = e^ãt sin(μt)
are a fundamental set of solutions and state the general solution.​

a) Brooke's monthly income is $3,166.67 from her full-time job as a cook, and an additional $532.59 from her part-time job, resulting in a total monthly income of $3,699.26.

b) Yes, the rent of $975 per month for the new apartment falls within the housing guideline of spending 30% or less of her monthly income on housing.

a) To determine Brooke's monthly income, we need to calculate her earnings from both her full-time and part-time jobs.

Full-time job income: Brooke earns $38,000 per year as a cook.

To find her monthly income from this job, we divide her annual salary by 12:

Monthly income from full-time job = $38,000 / 12 = $3,166.67

Part-time job income: Brooke earns $10.25 per hour and works 12 hours per week.

To find her weekly income from this job, we multiply her hourly rate by the number of hours she works:

Weekly income from part-time job = $10.25/hour x 12 hours/week = $123

To find her monthly income, we multiply her weekly income by the average number of weeks in a month (approximately 4.33):

Monthly income from part-time job = $123/week x 4.33 weeks/month = $532.59 (rounded to the nearest cent)

To calculate Brooke's total monthly income, we add her full-time and part-time job incomes:

Total monthly income = $3,166.67 + $532.59 = $3,699.26 (rounded to the nearest cent)

b) The rent for the new apartment is $975 per month, including utilities. To determine if it falls within the housing guidelines, we need to compare it to a percentage of Brooke's monthly income.

Typically, a common guideline is to spend no more than 30% of your monthly income on housing.

Percentage of monthly income for housing = 30% of total monthly income

= 30/100 x $3,699.26

= $1,109.78 (rounded to the nearest cent)

Since the rent of $975 is lower than the guideline of $1,109.78, if Brooke takes this apartment, it will fall within the housing guidelines.

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Answer:

-5/4

Step-by-step explanation:

Let [tex](x_1,y_1)=(-4,4)[/tex] and [tex](x_2,y_2)=(0,-1)[/tex]. The slope of the line would be:

[tex]\displaystyle \frac{y_2-y_1}{x_2-x_1}=\frac{-1-4}{0-(-4)}=\frac{-5}{4}=-\frac{5}{4}[/tex]

Answer: -5/4

Step-by-step explanation:

To find the slope between two points, you can use the formula:

Slope = (y2 - y1)/(x2 - x1)

Using the points (0, -1) and (-4, 4), we can substitute the coordinates into the formula:

slope = (4 - (-1))/(-4 - 0)

slope = (4 + 1)/(-4)

slope = 5/-4

Therefore, the slope between the two points is -5/4.

(a) To find the assets in 2011 using the given information: A. To find the assets in 2011, substitute 11 for x and evaluate to find A(x).

In 2011 the assets are about $669.6 billion

(b) To find the assets in 2016 using the given information: B. To find the assets in 2016, substitute 16 for x and evaluate to find A(x).

In 2016 the assets are about $931.5 billion.

(c) To find the assets in 2019 using the given information: B. To find the assets in 2019, substitute 19 for x and evaluate to find A(x).

In 2019 the assets are about $1135.4 billion.

Based on the information provided, we can logically deduce that the assets for a financial firm can be approximately represented by the following exponential function:

[tex]A(x)=324e^{0.066x}[/tex]

where:

For the year 2011, the cost (in billions of dollars) is given by;

x = (2011 - 2007) + 7

x = 4 + 7

x = 11 years.

Next, we would substitute 11 for x in the exponential function:

[tex]A(11)=324e^{0.066 \times 11}[/tex]

A(11) = $669.6 billions.

Part b.

For the year 2016, the cost (in billions of dollars) is given by;

x = (2016 - 2007) + 7

x = 9 + 7

x = 16 years.

Next, we would substitute 16 for x in the exponential function:

[tex]A(16)=324e^{0.066 \times 16}[/tex]

A(16) = $931.5 billions.

Part c.

For the year 2019, the cost (in billions of dollars) is given by;

x = (2019 - 2007) + 7

x = 12 + 7

x = 19 years.

Next, we would substitute 19 for x in the exponential function:

[tex]A(19)=324e^{0.066 \times 19}[/tex]

A(19) = $1135.4 billions.

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Answer:

56

Step-by-step explanation:

f(x)=4x^2-7

f'(x)=8x

f'(7)=56

Answer:

a)

[tex] \frac{1}{( - 7)^{4} } [/tex]

Answer:

[tex](-7)^-^4=\frac{1}{(-7)^4}[/tex]

Step-by-step explanation:

The user aswati already wrote the correct answer, but I wanted to help explain why their answer is correct so that you'll understand.

According to the negative exponent rule, when a base (let's call it m) is raised to a negative exponent (let's call it n), we rewrite it as a fraction where the numerator is 1 and the denominator is the base raised to the same exponent turned positive.

Thus, the negative exponent rule is given as:

[tex]b^-^n=\frac{1}{b^n}[/tex]

Thus, [tex](-7)^-^4[/tex] becomes [tex]\frac{1}{(-7)^4}[/tex]

Answer:

$6240

Step-by-step explanation:

given likelihoods:

sunny day = 14% = 0.14

cloudy day = 26% = 0.26

rainy day = 60% = 0.60

profits:

profit on a sunny day = $40,000

profit on a cloudy day = $14,000

Loss on a rainy day = -$5,000

expected profit = (probability of sunny day * profit on sunny day) + (probability of cloudy day * profit on cloudy day) + (probability of rainy day * loss on rainy)

expected profit = (0.14 * $40,000) + (0.26 * $14,000) + (0.60 * -$5,000)

=6240

The proposal is as follows

Part III  -  The distance between buoys A and B is 12.8 kilometers.

Part IV  -  The length of the other two triangle legs are 10.2 kilometers and 8.4 kilometers.

Part V  - The total length of the race course is 31.4 kilometers.

Part VIII  - The winner's speed during last year's race was 10.8 knots.

See the proposal attached.

Dear Sponsor,

I'm seeking sponsorship for the San Francisco sailboat race.

With a proven track record and the determination to win, your investment of $5,500 covers travel costs and potential hazards.

By associating your brand with a winning sailor, you'll gain significant exposure to thousands of spectators. Join me in this thrilling race for success.

Sincerely,

[Your Name]

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The measure of angle of arc UB is 46°

An arc is a part of the circumference.

If two arcs in circles with equal radii have the same length, then their central angles (and measures) will be equal.

To find the measure of angle of arc UB, we use the theorem that says;

angle I = 1/2( angle ZC - angle UB)

angle I = 48°

angle ZC = 142°

Therefore ;

represent angle UB by x

48 = 1/2( 142-x)

142 - x = 96

x = 142 -96

x = 46°

Therefore the measure of angle of arc UB is 46°

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Answer:

number is 16

Step-by-step explanation:

let n be the number , then 7 more than the number is n + 7 and 9 less than twice the number is 2n - 9

equating the 2 expressions

2n - 9 = n + 7 ( subtract n from both sides )

n - 9 = 7 ( add 9 to both sides )

n = 16

the required number is 16

The answer is:

Work/explanation:

Let's call the number n.

Seven more than n = n + 7

Nine less than twice n = 2n - 9

Put the expressions together :           n + 7 = 2n - 9

Now, we have an equation that we can solve for n.

First, flop the equation

2n - 9 = n + 7

Subtract n from each side

n - 9 = 7

Add 9 to each side

n = 16

Given the function `f(x)=1/x-1 -2` the task is to write the equation of the asymptote(s) of f and determine the x-intercept of f. Asymptotes are lines that the curve of a function approaches but never touches. There are two types of asymptotes: vertical and horizontal.

Vertical Asymptote Vertical asymptotes occur when a function approaches infinity or negative infinity at a specific value of x. This can occur in a rational function where there is a division by zero.

A vertical asymptote is found when the denominator of the rational function becomes zero. Since division by zero is undefined, it means that the rational function approaches infinity or negative infinity.

The equation of the vertical asymptote is x = a where a is the value that makes the denominator zero.

Horizontal AsymptoteA horizontal asymptote occurs when a function approaches a constant value (y) as x approaches infinity or negative infinity. A horizontal asymptote occurs when the degree of the numerator and denominator is the same.

The horizontal asymptote is found by comparing the degrees of the numerator and denominator and dividing the leading coefficient of the numerator by the leading coefficient of the denominator.3.1 Equation of the asymptotes of the equation of the vertical asymptote is x=1.

The degree of the numerator is less than the degree of the denominator. Therefore, the horizontal asymptote is y=0The equation of the horizontal asymptote is y=0.3.2 X-intercept of fTo find the x-intercept of f, set y=0.f(x) = 0= 1/(x-1) -2Add 2 to both sides2 = 1/(x-1)Take the reciprocal of both sides of the equation.1/2 = (x-1)x-1 = 2x = 3Hence, the x-intercept of the function is (3,0)

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The statement which is not true include the following: (2) AABM is isosceles.

In Mathematics and Geometry, an isosceles triangle simply refers to a type of triangle with two (2) sides that are equal in length and two (2) equal angles.

By critically observing circle M shown above, we can reasonably infer and logically deduce the following true statements:

ΔABC is a right triangle.

BM = MC (radius of circle M).

mBC = m∠BMC

mAB = 1/2m∠ACB (intersecting secant theorem).

In this context, we can logically conclude that triangle ABM does not represent an isosceles triangle because it does not have two (2) sides that are equal in length.

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Answer:

Step-by-step explanation:

To format and solve the equation "f(x) - 4" with the given function "f(x) = 2x + 1," we substitute the function into the equation and solve for x. Here's how it can be done:

f(x) - 4 = 2x + 1 - 4

Simplifying further:

f(x) - 4 = 2x - 3

To answer the question "f(4)" using the function f(x) = 2x + 1, we substitute x = 4 into the function:

f(4) = 2(4) + 1

Simplifying further:

f(4) = 8 + 1

f(4) = 9

Therefore, the value of f(4) is 9 when using the function f(x) = 2x + 1.

The angle m∠MHU between the intersection of the secant line HM and tangent line HU is equal to 54°

To calculate for the angle between the intersection of a secant and a tangent we need to know the measure of the intercepted arc, and then divide it by 2 to get the angle.

If the measure of the arc HM is given to be equal to 108°, then the measure of angle MHU is calculated as:

angle MHU = 108°/2

m∠MHU = 54°

Therefore, the angle m∠MHU between the intersection of the secant line HM and tangent line HU is equal to 54°

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The ordered pair given in the first row of the table can be written using function notation as (x, f(x)) = (-6, 8).

f(x) = -5 when x is 4.

In function notation, we represent the input value as 'x' and the corresponding output value as 'f(x)'.

Looking at the first row of the table, we see that when x is -6, the corresponding value of f(x) is 8.

Therefore, we can write this ordered pair as (-6, 8) in function notation.

Similarly, we can determine that f(x) = -5 when x is 4 by examining the second row of the table.

The value of f(x) is -5 when x is 4, so we can express it as f(4) = -5.

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The function f(x) = -0.5(x + 5)³(x + 3)(x - 3) satisfies the specified conditions of decreasing at x = -5, having a local minimum at x = -3, and a local maximum at x = 3.

One possible function that satisfies the given properties is:

f(x) = -0.5(x + 5)³(x + 3)(x - 3)

Check as follows:

Decreasing at x = -5:

Taking the derivative of f(x) and evaluating it at x = -5, we have:

f'(x) = -1.5(x + 5)²(x + 3)(x - 3) - 0.5(x + 5)³

f'(-5) = -1.5(0)²(-2)(-8) - 0.5(0)³ = 0 - 0 = 0

The derivative is zero at x = -5, therefore the function has a critical point at that location. To check if it is a maximum or minimum, we can examine the second derivative.

Taking the second derivative:

f''(x) = -3(x + 5)(x + 3)(x - 3) - 3(x + 5)²(x - 3)

f''(-5) = -3(0)(-2)(-8) - 3(0)²(-8) = 0 - 0 = 0

The second derivative is also zero at x = -5. However, since the first derivative is negative for x < -5 and positive for x > -5, this means that f(x) is decreasing at x = -5.

Local minimum at x = -3:

To check if f(x) has a local minimum at x = -3, we can examine the first and second derivatives at that point.

Taking the first derivative:

f'(-3) = -1.5(2)²(0)(-6) - 0.5(2)³ = 0

The first derivative is zero at x = -3, indicating a critical point.

Taking the second derivative:

f''(-3) = -3(2)(0)(-6) - 3(2)²(-6) = 0 - 72 = -72

Since the second derivative is negative at x = -3, this confirms the presence of a local minimum.

Local maximum at x = 3:

To check if f(x) has a local maximum at x = 3, we can again examine the first and second derivatives at that point.

Taking the first derivative:

f'(3) = -1.5(8)²(6)(0) - 0.5(8)³ = 0

The first derivative is zero at x = 3, indicating a critical point.

Taking the second derivative:

f''(3) = -3(8)(6)(0) - 3(8)²(0) = 0 - 0 = 0

The second derivative is zero at x = 3, indicating that the test is inconclusive. However, since the first derivative is positive for x < 3 and negative for x > 3, this means that f(x) is decreasing at x = 3.

Therefore, the function f(x) = -0.5(x + 5)³(x + 3)(x - 3) satisfies all the given conditions.

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Answer:

It's A and B: Opposite sides are parallel and Corresponding parts of congruent triangles are congruent.

Step-by-step explanation:

con un conjunto de deshielo determinante igual a 4, es x = 2.

Para determinar el valor de x en la matriz x 2

5 7

dado que el conjunto de deshielo determinante es igual a 4, necesitamos utilizar la propiedad de que el determinante de una matriz 2x2 se puede calcular utilizando la siguiente formula:

determinante = (a × d) - (b × c)

Donde a, b, c, y d son los elementos de la matriz.

En este caso, tenemos la matriz:

x 2

5 7

Aplicando la formula del determinante, podemos establecer la siguiente ecuacion:

( x × 7 ) - ( 2 × 5 ) = 4

Simplificando la ecuacion, obtenemos:

7x - 10 = 4

A continuacion, vamos a resolver la ecuacion para encontrar el valor de x:

7x = 4 + 10

7x = 14

Dividiendo ambos lados de la ecuacion por 7, obtenemos:

x = 2

Por lo tanto, el valor de x en la matriz x 2

5 7

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Answer:

8

Step-by-step explanation:

Answer:

The volume of the cylinder is 628.318530718

Step-by-step explanation:

The formula used to find the volume of a cylinder (v) is [tex]v = \pi r^2h[/tex], where r = radius and h = height. As the question says to keep the answer exact, we will be using pi as opposed to 3.14.

The radius is 5, and the height is 8. Plug these values into the equation and solve:

[tex]v =\pi *5^2*8[/tex]

[tex]v = 628.318530718[/tex]

So, the exact volume of the cylinder is 628.318530718. Rounded is 628.32

Answer:

200π or 628

Step-by-step explanation:

Note: your picture is not clear so I am assuming the height to be 8.

r = 5

h = 8

Volume = πr²h

= π * 5² * 8

= (25*8) π

= 200π

= 200*3.14

= 628

Step-by-step explanation:

Common difference , d, is  9

Sn = n/2 ( a1 + a33)                a33 = a1 + 32d = 2 + 32(9) = 290

S33 = 33/2 ( 2+290) = 4818

The data set has two outliers, namely the points (2, 225) and (8, 75).

Based on the given information about the scatterplot, we can observe that most of the points are grouped closely together and show a slight increase.

There are two points that lie outside of this cluster, specifically (2, 225) and (8, 75).

To determine if these points are outliers, we need to consider their deviation from the general pattern exhibited by the majority of the data points.

If these points deviate significantly from the overall trend, they can be considered outliers.

In this case, since (2, 225) and (8, 75) lie outside of the cluster of closely grouped points and do not follow the general pattern, they can be considered outliers.

These points are noticeably different from the majority of the data points and may have influenced the overall trend of the scatterplot.

The data set has two outliers, namely the points (2, 225) and (8, 75).

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The distance from point A to point B is approximately 16.64 units. None of the given options (A, B, C, D) match this value exactly, so there seems to be an error in the options provided.

To find the distance from point A to point B, we can use the distance formula in Euclidean geometry. The distance formula between two points (x1, y1) and (x2, y2) is given by:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, point A is (-8, -3) and point B is (6, 6). Plugging these values into the distance formula, we have:

distance = sqrt((6 - (-8))^2 + (6 - (-3))^2)

= sqrt((6 + 8)^2 + (6 + 3)^2)

= sqrt(14^2 + 9^2)

= sqrt(196 + 81)

= sqrt(277)

≈ 16.64

Thus, the distance between points A and B is roughly 16.64 units. Since none of the available options (A, B, C, or D) exactly match this value, there appears to be a problem with the options.

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The total petty cash expenditures would be =$130.84

To calculate the petty cash expenditures, the following is added up as follows;

The cost for stamp = $12.50

The cost for coffee supplies = $25.19

The cost for pizza delivery = $15.50

The cost for white board markers = $20.00

The cost for sympathy greeting card= $5.78

The cost of flowers for Jean's retirement farewell = $39.87

The cost of courier = $12.00

Therefore the total petty cash expenditures would be= 12.50+25.19+15.50+20+5.78+39.87+12= $130.84

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a) The least squares regression line for the given data is y = -3.5358x + 87.2857.

b) The slope of -3.5358 indicates that for each additional day absent, the final grade is expected to decrease by approximately 3.5358 points on average.

c) The y-intercept of 87.2857 represents the estimated final grade when the number of absences is zero, implying that a student who did not miss any classes is expected to have a final grade of approximately 87.2857.

a) To find the least squares regression line, we need to determine the equation of the line in the form y = mx + b, where m represents the slope and b represents the y-intercept.

Let's calculate the necessary sums:

∑x = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

∑y = 89.2 + 86.4 + 83.5 + 81.1 + 78.2 + 73.9 + 64.3 + 71.8 + 65.5 + 66.2 = 759.9

∑xy = (0 * 89.2) + (1 * 86.4) + (2 * 83.5) + (3 * 81.1) + (4 * 78.2) + (5 * 73.9) + (6 * 64.3) + (7 * 71.8) + (8 * 65.5) + (9 * 66.2) = 5079.6

∑[tex]x^2[/tex] = [tex](0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) + (6^2) + (7^2) + (8^2) + (9^2)[/tex] = 285

[tex]\sum y^2 = (89.2^2) + (86.4^2) + (83.5^2) + (81.1^2) + (78.2^2) + (73.9^2) + (64.3^2) + (71.8^2) + (65.5^2) + (66.2^2) = 59718.63[/tex]

Using the formulas for the slope (m) and y-intercept (b):

m = (n∑xy - (∑x)(∑y)) / (n∑x^2 - (∑x)^2)

b = (∑y - m(∑x)) / n

Substituting the calculated values:

m = (10 * 5079.6 - (45 * 759.9)) / (10 * 285 - (45)^2)

b = (759.9 - m(45)) / 10

Calculating the values:

m ≈ -3.5358

b ≈ 87.2857

Therefore, the least squares regression line is y = -3.5358x + 87.2857.

b) The slope (-3.5358) represents the change in the final grade (y) for each additional day absent (x). This means that on average, for each additional day a student is absent, their final grade is expected to decrease by approximately 3.5358 points.

c) The y-intercept (87.2857) represents the estimated final grade when the number of absences (x) is zero. In other words, it is the predicted final grade for a student who did not miss any classes.

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Question

The following data represents the number of days absent, x, and the final grade, y, for a sample of college students at a community college.

No.of absences, x | 0 1 2 3 4 5 6 7 8 9

Final Grade, y | 89.2 86.4 83.5 81.1 78.2 73.9 64.3 71.8 65.5 66.2

a) Find the least squares regression line treating the number of absences as the explanatory variable and the final grade as the response variable.

b) Interpret the slope using complete sentences.

c) Interpret the y intercept using complete sentences.

Answer:

Step-by-step explanation:

To determine the length of AB, we need to find the value of x.

We are given that AC = 2x + 7, BC = x, and AB = 5x + 9.

Since B is on the line segment AC, the sum of lengths AB and BC should equal the length of AC. Therefore, we can set up the equation:

AB + BC = AC

Substituting the given values, we have:

(5x + 9) + x = 2x + 7

Simplifying the equation:

6x + 9 = 2x + 7

Bringing like terms to one side:

6x - 2x = 7 - 9

4x = -2

Dividing both sides by 4:

x = -2/4

Simplifying:

x = -1/2

Now that we have the value of x, we can substitute it back into the expression for AB to find its numerical length:

AB = 5x + 9 = 5(-1/2) + 9 = -5/2 + 9 = (18 - 5)/2 = 13/2 = 6.5

Therefore, the numerical length of AB is 6.5.

1. the linear equation 5(x + 7) - 3(x - 4) = 7x + 2, x = 9

2. In linear equation 4(3x + 5) - 3 = 9x - 7, x = -8

3. the linear equation 1/3(5x - 9) = 2(1/3x + 6), x = 15

A linear equation is an equation in only on variable.

1. To solve the linear equation 5(x + 7) - 3(x - 4) = 7x + 2, we proceed as follows

5(x + 7) - 3(x - 4) = 7x + 2

Expanding the brackets, we have

5x + 35 - 3x + 12 = 7x + 2

Collecting like terms in the expression, we have

5x + 35 - 3x + 12 = 7x + 2

5x - 3x - 7x = 2 - 35 - 12

-5x = -45

x = -45/-5

x = 9

2. To solve linear equation 4(3x + 5) - 3 = 9x - 7, we proceed as follows

4(3x + 5) - 3 = 9x - 7

Expanding the brackets, we have

12x + 20 - 3 = 9x - 7

Collecting like terms in the expression, we have

12x - 9x = - 7 + 3 - 20

3x = -24

x = -24/3

x = -8

3. To solve  the linear equation 1/3(5x - 9) = 2(1/3x + 6), we proceed as follows

1/3(5x - 9) = 2(1/3x + 6)

Expanding the brackets, we have

5x/3 - 3 = 2/3x + 12

Collecting like terms, we have

5x/3 - 2x/3 = 12 + 3

3x/3 = 15

x = 15

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So, the complex number equivalent to the given expression is 0 + 9i, which can also be written as 9i.

To simplify the expression 1/3(6 + 3i) - 2/3(6 - 12i), we can perform the necessary calculations.

First, let's simplify each term separately:

1/3(6 + 3i) = 2 + i (divide each term by 3)

2/3(6 - 12i) = 4 - 8i (divide each term by 3)

Now, let's substitute these simplified terms back into the original expression:

2 + i - (4 - 8i)

When subtracting complex numbers, we distribute the negative sign:

2 + i - 4 + 8i

Combine like terms:

(-2 + 2) + (i + 8i) = 0 + 9i

The expression 1/3(6 + 3i) - 2/3(6 - 12i) simplifies to 9i.

We can make the necessary computations to simplify the statement 1/3(6 + 3i) - 2/3(6 - 12i).

Let's first simplify each phrase individually:

Divide each term by 3 to get 1/3(6 + 3i) = 2 + i.

Divide each term by 3 to get 2/3(6 - 12i) = 4 - 8i.

Let's now add these abbreviated terms back into the original phrase:

2 + i - (4 - 8i)

Distributing the negative sign while subtracting complex numbers is as follows:

2 + i - 4 + 8i

combining similar terms

(-2 + 2) + (i + 8i) = 0 + 9i

A simplified version of the phrase 1/3(6 + 3i) - 2/3(6 - 12i) is 9i.

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