(2 – 3i) and (4 + 3i)
The product of the two complex numbers above is a - 6i, where a is a constant. What is the value of a ?
(Note: i = -1)

This looks like a system of differential equations.

[tex]\begin{cases}x' + 2y' + x = 0 \\ x' - y' + y = 0 \\x(0)=0, y(0)=1\end{cases}[/tex]

Eliminating x' gives

[tex](x'+2y'+x)-(x'-y'+y) = 0 - 0 \implies 3y' - y + x = 0[/tex]

and eliminating y' gives

[tex](x'+2y'+x) + 2(x'-y'+y)=0+2\times0 \implies 3x' + x + 2y = 0[/tex]

so that we can rewrite the system as

[tex]\begin{cases}3x' = -x - 2y \\ 3y' = -x + y\end{cases}[/tex]

or equivalently in matrix form as

[tex]\begin{bmatrix}x\\y\end{bmatrix}' = \dfrac13 \begin{bmatrix}-1&-2\\-1&1\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}[/tex]

Compute the eigenvalues for the coefficient matrix:

[tex]\det\begin{bmatrix}-1-\lambda&-2\\-1&1-\lambda\end{bmatrix} = (-1-\lambda)(1-\lambda) - 2 = \lambda^2 -3 = 0 \implies \lambda=\pm\sqrt3[/tex]

Compute the corresponding eigenvectors:

[tex]\lambda=\sqrt3 \implies \begin{bmatrix}-1-\sqrt3&-2\\-1&1-\sqrt3\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix} \\\\ \implies v_1=(1-\sqrt3)v_2 \implies \vec v = \begin{bmatrix}1-\sqrt3\\1\end{bmatrix}[/tex]

[tex]\lambda=-\sqrt3 \implies \begin{bmatrix}-1+\sqrt3&-2\\-1&1+\sqrt3\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix} \\\\ \implies v_1 = (1+\sqrt3)v_2 \implies \vec v = \begin{bmatrix}1+\sqrt3\\1\end{bmatrix}[/tex]

We end up multiplying the matrix by 1/3, so the eigenvalues also get scaled by 1/3 and λ = ±1/√3. The eigenvectors stay the same.

Then the characteristic solution to the system is

[tex]\begin{bmatrix}x\\y\end{bmatrix} = C_1 e^{t/\sqrt3} \begin{bmatrix}\frac{1-\sqrt3}3\\\frac13\end{bmatrix} + C_2 e^{-t/\sqrt3} \begin{bmatrix}\frac{1+\sqrt3}3\\\frac13\end{bmatrix}[/tex]

Use the initial conditions to solve for the constants.

[tex]\begin{bmatrix}0\\1\end{bmatrix} = C_1 \begin{bmatrix}\frac{1-\sqrt3}3\\\frac13\end{bmatrix} + C_2 \begin{bmatrix}\frac{1+\sqrt3}3\\\frac13\end{bmatrix}[/tex]

[tex]\implies \begin{cases}(1-\sqrt3) C_1 + (1+\sqrt3) C_2 = 0 \\ C_1 + C_2 = 3\end{cases}[/tex]

[tex]\implies C_1=\dfrac{3+\sqrt3}2, C_2=\dfrac{3-\sqrt3}2[/tex]

Then the particular solution is

[tex]\begin{bmatrix}x\\y\end{bmatrix} = \dfrac{3+\sqrt3}6 e^{t/\sqrt3} \begin{bmatrix}1-\sqrt3\\1\end{bmatrix} + \dfrac{3-\sqrt3}6 e^{-t/\sqrt3} \begin{bmatrix}1+\sqrt3\\1\end{bmatrix}[/tex]

or

[tex]\begin{cases}x(t) = -\dfrac1{\sqrt3} e^{\sqrt3 \, t} + \dfrac1{\sqrt3} e^{-\sqrt3\,t} \\\\ y(t) = \dfrac{3+\sqrt3}6 e^{\sqrt3\,t} + \dfrac{3-\sqrt3}6 e^{-\sqrt3\,t}\end{cases}[/tex]

Answer:

72.5% out of 100%

Step-by-step explanation:

0.725 is 72.5% of 1. 100% technically counts as 1. so 72.5%

Answer:

400

Step-by-step explanation:

Division cancels multiplication. Divide by 0.17 to isolate "x" on the right side of the equation.

68 = 0.17 * x

divide both sides by 0.17

400 = x

68 is 17 percentage of the number 400.

To find the number that 68 is 17% of, we can set up the equation:

17% of x = 68

To solve for x, we need to divide both sides of the equation by 17% (or 0.17 as a decimal):

x = 68 / 0.17

Evaluating the division:

x=400

Therefore, 68 is 17% of 400.

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Answer:

-7

Step-by-step explanation:

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Answer:

-7

Step-by-step explanation:

Answer:

x = 2     y = -12

Step-by-step explanation:

x+ y = -10

2 + y = -10

      y = -10 -2 = -12

Answer: Heat pollution.

If you are to search up

heat pollution occurs because?

the answer that will pop up is:

The main reasons of thermal pollution are industries and power plants, withdrawing water (a powerful sink for heat waste) from streams for cooling their generators and machines, and then releasing back into the water bodies, which elevates the temperature of freshwater habitats (Miara et al., 2018).

Answer:

Step-by-step explanation:

Total paycheck before taking out taxes and insurance :

38×15 = 570

Left Amount of money after applying taxes :

570 - 570×(p/100)

= 570×[1 - (p/100)]

= 570×[(100-p)/100]

= (570/100) × [100 - p]

= 5.7 × [100 - p]

After taking out  the insurance amounts we can write this equation:

.,7 × [100 - p] - (35+4) = 462.6

⇔ 5.7 × [100 - p] - 39 = 462.6

⇔ 5,7 × [100 - p] = 462.6 + 39

⇔ 5,7 × [100 - p] = 501.6

⇔ 100 - p = 501.6/5.7

⇔ 100 - p = 88

⇔ p = 100 - 88

⇔ p = 12

Answer:

the answer for this one it should be the probability or chances that the event b occurs when the event a has occured.

Step-by-step explanation: I hope this help you pal or folk and have a great day.

Answer:

Now plot those points in graph and find intersection hope it helps you.......

to get the equation of any straight line, we simply need two points off of it, hmmm let's use (0 , 10) and (8 , 30) from the table

[tex](\stackrel{x_1}{0}~,~\stackrel{y_1}{10})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{30}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{30}-\stackrel{y1}{10}}}{\underset{run} {\underset{x_2}{8}-\underset{x_1}{0}}}\implies \cfrac{20}{8}\implies \cfrac{5}{2}[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{10}=\stackrel{m}{\cfrac{5}{2}}(x-\stackrel{x_1}{0}) \\\\\\ y-10=\cfrac{5}{2}x\implies y=\cfrac{5}{2}x+10[/tex]

Answer: C

Step-by-step explanation:

[tex]5\left(4x+5\right)[/tex]

[tex]\mathrm{Apply\:the\:distributive\:property}:\quad \:a\left(b+c\right)=ab+ac[/tex]

[tex]=5\cdot \:4x+5\cdot \:5[/tex]

[tex]\bf=20x+25[/tex]

Answer: -310

Step-by-step explanation:

-1240÷4

Answer:

  $7045.82

Step-by-step explanation:

The formula for the balance of an account earning interest at rate r compounded continuously for t years is ...

  A = Pe^(rt)

We want to find P, given the other parameters in this formula.

__

  38956.06 = P(e^(0.09×19)) = P(5.52896148)

  P = 38956.06/5.52896148 ≈ 7045.82

The amount of the initial investment was about $7045.82.

Answer:D

Step-by-step explanation: Multiply the expression

To find the missing angle's measure

 ⇒ important to understand the sum of the angles of a triangle

    ⇒ the sum of every angle of any triangle in all cases equal to 180

         degrees

What is the information given:

 ⇒ one angle is of 94 degrees and the other of 46 degrees

Using the information in the first section and the information given in the diagram, let's set up an equation:

  [tex]94+46+x = 180\\140+x = 180\\x = 40[/tex]

The measure of the missing angle is 40 degrees

Hope that helps!

The rational function shown in the graph is:

[tex]y = \frac{x - 6}{x - 5}[/tex]

First, we can see that we have an asymptote at x = 5, so the denominator becomes equal to zero when x = 5.

Then the rational function is something like:

[tex]y = \frac{g(x)}{x - 5}[/tex]

We also can see that when x = 6, the function intercepts the x-axis, then we must have that:

g(6) = 0

So we can define:

g(x) = x - 6

So the rational function is:

[tex]y = \frac{x - 6}{x - 5}[/tex]

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Answer: i think the answer is 116. if it is not i am terribly sorry.

Step-by-step explanation:

Answer: I believe B and F

Step-by-step explanation:

Answer:

152 cups

Step-by-step explanation:

1. 44 quarts are in 11 gallons.

2. 44-6=38.

3. There are 152 cups in 38 quarts.

Answer:

Edna had $510 more amount of money than Toby in the end.

Step-by-step explanation:

LET :-

Edna's amount of money - x

Fatimah's amount of money - y

Toby's amount of money - z

Now,

By the question,

x = 4y

y : (y + z) = 5 : 8

y - 30 = z + 30

Now,

y : (y + z) = 5 :8

or, y/(y + z) = 5/8

or, 8y = 5y + 5z

or, 3y = 5z

so, y = 5z/3

Now,

y - 30 = z + 30

or, 5z/3 - 30 = z + 30

or, (5z - 90)/3 = z + 30

or, 5z - 90 = 3z + 90

or, 5z - 3z = 90 + 90

or, 2z = 180

so, z = 90

Toby's amount of money - $90

Now,

y = 5z/3

or, y = 5 * 90/3

or, y = 5 * 30

so, y = 150

Fatimah's amount of money - $150

Again,

x = 4y

or, x = 4*150

or, x = 600

Edna's amount of money - $600

Now,

$600 - $90

= $510

Therefore,Edna had $510 more amount of money than Toby in the end.

Answer:

The answer is 3x.

Step-by-step explanation:

I used a calculator

Step-by-step explanation:

the word kilo means thousand. By that logic we know that 2200 grams are in 2.2 kilograms

Answer:

vertex = (- 5, - 13 )

Step-by-step explanation:

given a quadratic in standard form : y = ax² + bx + c ( a ≠ 0 )

then the x- coordinate of the turning point ( vertex) is

x = - [tex]\frac{b}{2a}[/tex]

y = x² + 10x + 12 ← is in standard form

with a = 1, b = 10 , then

x = - [tex]\frac{10}{2}[/tex] = - 5

substitute x = - 5 into the equation and evaluate for y

y = (- 5)² + 10(- 5) + 12 = 25 - 50 + 12 = - 13

turning point = (- 5, - 13 )

Answer:

Give two equivalent ratios of 6 : 4. 6 : 4 = 6/4 So, equivalent ratios are 6/4 × 2/2 = 12/8 = 12:8 6/4 × 3/3 = 18/12 = 18:12 So, 6 : 4 = 12 : 8 = 18 : 12

Step-by-step explanation:

Using proportions, it is found that the total purchase price of the bracelet is of $10,769.

A proportion is a fraction of a total amount.

In this problem, we consider the two taxes, of 6% and 11%, hence the price suffers an increase of 17%, that is, the price is multiplied by 117% = 1.17 to reach $12,600, hence:

1.17p = 12600

p = 12600/1.17

p = 10769.

The total purchase price of the bracelet is of $10,769.

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