x+2y=6
-7x+3y=-8 (using substitution)

Answer:

Step-by-step explanation:

the Laplace transform of the function f(x) = e^(ax) is 1/(a-s).

The definition of the Laplace transform of a function f(t) is given by:

L{f(t)} = F(s) = ∫_0^∞ e^(-st) f(t) dt

where s is a complex number.

If f(x) = 0, then we have:

L{f(x)} = L{0} = ∫_0^∞ e^(-st) 0 dt = 0

Therefore, the Laplace transform of the zero function is zero.

If f(x) = 1, then we have:

L{f(x)} = L{1} = ∫_0^∞ e^(-st) dt

Using integration by parts, we get:

L{1} = ∫_0^∞ e^(-st) dt = [-e^(-st)/s]_0^∞ = [0 - (-1/s)] = 1/s

Therefore, the Laplace transform of the constant function 1 is 1/s.

If f(x) = x, then we have:

L{f(x)} = L{x} = ∫_0^∞ e^(-st) x dt

Using integration by parts again, we get:

L{x} = ∫_0^∞ e^(-st) x dt = [(-e^(-st) x)/s]_0^∞ + (1/s) ∫_0^∞ e^(-st) dt

Since e^(-st) x approaches zero as t approaches infinity, the first term evaluates to zero. We can then simplify the second term using the result from part 2:

L{x} = (1/s) ∫_0^∞ e^(-st) dt = 1/s * (1/s) = 1/s^2

Therefore, the Laplace transform of the function f(x) = x is 1/s^2.

If f(x) = e^(ax), then we have:

L{f(x)} = L{e^(ax)} = ∫_0^∞ e^(-st) e^(ax) dt

Simplifying the integrand, we get:

L{e^(ax)} = ∫_0^∞ e^((a-s)t) dt

We can evaluate this integral using the formula:

∫_0^∞ e^(-bx) dx = 1/b

Setting b = a - s, we get:

L{e^(ax)} = ∫_0^∞ e^((a-s)t) dt = 1/(a-s)

Therefore, the Laplace transform of the function f(x) = e^(ax) is 1/(a-s).

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1465 will be the mean monthly salary .The answer is (A) Rs. 1465.

Let the sum of the 12 employees' salaries be S.

Then, the mean monthly salary of the 12 employees is given by:

S/12 = 1450

S = 12 * 1450

S = 17400

If one more person joins with a salary of Rs. 1645, the new sum of the 13 employees' salaries is:

S' = S + 1645

S' = 17400 + 1645

S' = 19045

The new mean monthly salary of the 13 employees is:

S'/13 = 19045/13

S'/13 = 1465

Therefore, the answer is (A) Rs. 1465.

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A graph of the points that represent this data are shown on the coordinate plane attached below.

In this scenario, the week number would be plotted on the x-axis (x-coordinate) of the scatter plot while the amount of money (in dollars) would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the week number and the amount of money (in dollars), a linear equation for the line of best fit is as follows:

y = 1.19x + 1.05

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The domain of function is the set of all possible values of x for which the function is defined. In this case, the function G(x) = 5 - 2x is defined for all real numbers of x.

The collection of all feasible input values (x-values) for which a function may be defined is known as the domain of function. In other words, it is the collection of all x values that may be passed into the function and provide a legitimate result (a y-value).

The function G(x) = 5 - 2x in this instance is a linear function, meaning it is defined for all real values of x. This is so that we may plug in any real value for x, and the function will output a real number for G(x) that corresponds. As an illustration, when we enter x = 0, we obtain G(0) = 5 - 2(0) = 5, and when we enter x = 2, we obtain G(2) = 5 - 2(2) = 1.

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The cost of 100 donuts is $ 1 if a dozen of donuts cost 12 cents.

This question is solved using the unitary method. The unitary method is a method in which you find the value of a unit and then the value of the required number of units.

1 dozen refers to a group of 12.

Cost of 1 dozen donuts or 12 donuts = 12 cents

Cost of 1 donut = [tex]\frac{12}{12}[/tex] = 1 cent

Cost of 100 donuts = 1 * 100 = 100 cents

100 cents = 1 dollar.

Thus, the cost of 100 donuts is 100 cents or 1 dollar.

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La probabilidad de que no haya demoras en un viaje de ida y vuelta es 0.64 (64%).

La probabilidad de que no haya demoras en un viaje de ida y vuelta se puede calcular utilizando la probabilidad complementaria. Si la probabilidad de que un vuelo se retrase es 0.2 (20%), entonces la probabilidad de que no haya retrasos en un vuelo individual es 1 - 0.2 = 0.8 (80%).

Para un viaje de ida y vuelta, la probabilidad de que no haya retrasos en ambos vuelos se calcula multiplicando las probabilidades de no retraso de cada vuelo.

Entonces, la probabilidad de que no haya demoras en un viaje de ida y vuelta sería 0.8 * 0.8 = 0.64 (64%), o 64 de cada 100 viajes de ida y vuelta no experimentarían retrasos.

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Answer:

2.25

Step-by-step explanation:

45 nickels

45*5=225

225 cents

2.25

The value of 45 nickels in decimal number can be 2.25.

In the decimal system, each digit's value depends on its position or place value within the number.

A nickel is worth 0.05 dollars.

To find the value of 45 nickels, multiply the number of nickels by the value of each nickel:

So, Value = Number of nickels × Value of each nickel

                = 45 × 0.05

                = 2.25

Therefore, the value of 45 nickels is $2.25 as a decimal number.

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The different teams of 6 students that can be formed for competitions is 3003

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

Students = 14

Students in the team = 6

Using the above as a guide, we have the following:

n = 14

r = 6

The different teams of 6 students that can be formed for competitions is calculated as

Teams = nCr

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

Teams = 14C6

So, we have

Teams = 14!/(6! * 8!)

Evaluate

Teams = 3003

Hence, the different teams of 6 students that can be formed for competitions is 3003

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Answer:

2

Step-by-step explanation:

The K and N are the trapeziums and the two lines opposite to them go in a parallel line

The images of points B and C are B'(x, y) = (- 2, 6) and C'(x, y) = (- 1, 7), respectively.

In this problem we find must determine the image of two points by translation, whose formula is introduced below:

T(x, y) = P'(x, y) - P(x, y)

Where:

First, determine the translation vector:

T(x, y) = (1, 4) - (0, 0)

T(x, y) = (1, 4)

Second, determine the images of points B and C:

B'(x, y) = (- 3, 2) + (1, 4)

B'(x, y) = (- 2, 6)

C'(x, y) = (- 2, 3) + (1, 4)

C'(x, y) = (- 1, 7)

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Answer:

108y^10x^17

Step-by-step explanation:

Answer: 3

Step-by-step explanation:

Answer:

2hrs 24 mins

Step-by-step explanation:

Ok so let's make this problem a bit simpler by splitting it up.

Tim paints a room in 6 hours.

So, we can also say that she paints 1/6 of that room in 1 hour

Bella paints it in 4 hours

So, we can also say that she paints 1/4 of that room in 1 hour

Now, lets see what we have:

Bella: 1/4 every hour

Tim: 1/6 every hour

The problem states that they are working together, so we need to add the values we have:

1/4 + 1/6

We cannot just add them, we must make them have the same common denominator.

LCD is 12, you can find that by just doing the times tables for 4 and 6 and seeing what number they match on first.

3/12 + 2/12 = 5/12

So, tim and bella working together paint 5/12 of a room in 1 hour.

They paint 5/12 of a room in 60 minutes

They paint 1/12 of the room in 12 minutes(divide both values by 5)

So if they paint 1/12 of the room in 12 minutes, we can multiply both values by 12 to get our answer.

They paint the full room in 144 minutes(12*12).

144 minutes is 2 hours and 24 minutes

Answer:

52.5

Step-by-step explanation:

15×7=105÷2

=52.5 ans it means fifteen times seven divided by two

Number of cars that coasted more than 10.75 feet = 1

To solve this problem, you need to compare the distance each car coasted to 10.75 feet, which is the threshold for determining whether a car coasted more or less than 10.75 feet.

Manuel's car coasted 10 feet, which is less than 10.75 feet, so it did not coast more than 10.75 feet.

Richard's car coasted 10.5 feet, which is also less than 10.75 feet, so it did not coast more than 10.75 feet either.

Diego's car coasted 102 feet, which is more than 10.75 feet. Therefore, only one car coasted more than 10.75 feet, and the answer is 1.

So the answer is:

Number of cars that coasted more than 10.75 feet = 1

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For the expressions  3x+4 –2x, and 5x+2x+x the simplified expression after combining like terms is x+4 and 8x.

The given expressions are  3x+4 –2x, and 5x+2x+x

We have to simplify these expressions by combining the like terms

For the expression 3x+4 –2x

We have to combine like terms

x+4

Now for expression  5x+2x+x

Combine the like terms to get

8x

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Answer: It's correct

Step-by-step explanation:

Given the function y = 4x² + 3x, we will find dy/dt by differentiating y with respect to t. Therefore, the value of dy/dt is -15.

Using the chain rule, we have:

dy/dt = (dy/dx)(dx/dt)

Differentiating y with respect to x, we get:

dy/dx = 8x + 3

Now, we are given that x = -1 and dx/dt = 3. We can substitute these values into our equation:

dy/dt = (8(-1) + 3)(3)

dy/dt = (-5)(3)

dy/dt = -15

So, when x = -1 and dx/dt = 3, the value of dy/dt is -15.

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(a) y(x) = c1 e^(-4x/3) cos(2x) + c2 e^(-4x/3) sin(2x), stable node at the origin;

(b) y(x) = c1 e^(2x) + c2 e^(-4x), unstable node at the origin;

(c) y(x) = c1 e^(-x/22) cos(sqrt(119)x/22) + c2 e^(-x/22) sin(sqrt(119)x/22), stable node at the origin;

(d) y(x) = c1 e^(sqrt(3)x/2) + c2 e^(-sqrt(3)x/2), unstable saddle at the origin;

(e) y(x) = c1 e^x cos(2x) + c2 e^x sin(2x), stable spiral at the origin;

(f) y(x) = c1 cos(3x/2) + c2 sin(3x/2), stable limit cycle around the origin;

(g) y(x) = c1 e^(-x/3) + c2 e^(-x), stable node at the origin.

(a) The characteristic equation is 2r^2 + 9r + 4 = 0, with roots r1 = -4/3 and r2 = -1/2. The general solution is y(x) = c1 e^(-4x/3) cos(2x) + c2 e^(-4x/3) sin(2x). The equilibrium at the origin is a stable node since both eigenvalues have negative real parts.

(b) The characteristic equation is r^2 + 2r - 8 = 0, with roots r1 = 2 and r2 = -4. The general solution is y(x) = c1 e^(2x) + c2 e^(-4x). The equilibrium at the origin is an unstable node since both eigenvalues have positive real parts.

(c) The characteristic equation is 44r^2 - 12r + 5 = 0, with roots r1 = (3 + sqrt(119))/22 and r2 = (3 - sqrt(119))/22. The general solution is y(x) = c1 e^(-x/22) cos(sqrt(119)x/22) + c2 e^(-x/22) sin(sqrt(119)x/22). The equilibrium at the origin is a stable node since both eigenvalues have negative real parts.

(d) The characteristic equation is 2r^2 - 3 = 0, with roots r1 = sqrt(3)/2 and r2 = -sqrt(3)/2. The general solution is y(x) = c1 e^(sqrt(3)x/2) + c2 e^(-sqrt(3)x/2). The equilibrium at the origin is an unstable saddle since the eigenvalues have opposite signs.

(e) The characteristic equation is r^2 - 2r + 5 = 0, with roots r1 = 1 + 2i and r2 = 1 - 2i. The general solution is y(x) = c1 e^x cos(2x) + c2 e^x sin(2x). The equilibrium at the origin is a stable spiral since both eigenvalues have negative real parts and non-zero imaginary parts.

(f) The characteristic equation is 4r^2 + 9 = 0, with roots r1 = 3i/2 and r2 = -3i/2. The general solution y(x) = c1 cos(3x/2) + c2 sin(3x/2), stable limit cycle around the origin.

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The correct choice is: C. The absolute maximum is at x = 1050, and there is no absolute minimum.

To find the absolute extrema of the function f(x) = -0.002x^2 + 4.2x - 50 over the interval (-∞, ∞), we need to find the critical points and then determine if there's a maximum or minimum at each point.

Step 1: Find the derivative of the function f(x) with respect to x. f'(x) = -0.004x + 4.2

Step 2: Set the derivative equal to zero and solve for x. -0.004x + 4.2 = 0 x = 1050

Step 3: Since we have only one critical point, we need to determine if it's a maximum or a minimum. To do this, we can use the second derivative test.

Step 4: Find the second derivative of the function f(x) with respect to x. f''(x) = -0.004

Step 5: Since the second derivative is negative (f''(x) = -0.004 < 0), the critical point x = 1050 corresponds to an absolute maximum. Step 6: Calculate the value of the function f(x) at x = 1050. f(1050) = -0.002(1050)^2 + 4.2(1050) - 50 = 2150

Thus, the absolute maximum is at x = 1050, and the value is 2150. Since the function is a parabola with the "mouth" facing downwards, there is no absolute minimum.

The correct choice is: C. The absolute maximum is at x = 1050, and there is no absolute minimum.

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Josue and Pho brought 3 1/5 liters of lemonade altogether

Josue and Pho brought 1 3/5 liters of lemonade each, so the total amount of lemonade they brought is:

1 3/5 + 1 3/5 = 3 1/5

To add the two mixed numbers, we first need to find a common denominator. In this case, the common denominator is 5. Then we convert both mixed numbers into fractions with a denominator of 5:

1 3/5 = (5 × 1 + 3) / 5 = 8/5

1 3/5 = (5 × 1 + 3) / 5 = 8/5

Now we can add the fractions:

8/5 + 8/5 = (8 + 8) / 5 = 16/5

Finally, we can convert the fraction back to a mixed number:

16/5 = 3 1/5

Therefore, Josue and Pho brought 3 1/5 liters of lemonade altogether.

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Answer:

Sure, here's the solution to the word problem:

Jack has $10 in his lunch account and plans to spend $2 a week on snacks. To find out how long it will take his lunch account to reach zero, we can divide the total amount of money in his account by the amount he spends each week.

```

$10 / $2 = 5 weeks

```

Therefore, it will take Jack 5 weeks to spend all of the money in his lunch account.

Here's another way to solve the problem:

We can also set up an equation to represent the situation. Let x be the number of weeks it takes Jack's lunch account to reach zero. We know that Jack starts with $10 and spends $2 each week, so we can write the equation:

```

$10 - $2x = 0

```

Solving for x, we get:

```

x = 5

```

Therefore, it will take Jack 5 weeks to spend all of the money in his lunch account.

Answer:

in 5 weeks he will have 0$ in his account

Step-by-step explanation:

The theoretical probability of the spinner not landing on yellow, would be 75 %.

In order to calculate the likelihood of the spinner not landing on yellow, it is necessary to initially identify the quantity of non-yellow partitions and subsequently divide this by the full tally of sections. The spinner comprises a total of 8 individual segments.

Of these, two (i.e., sections 2 and 3) are colored in shades of yellow, hence totaling two yellow sectors. This leaves a further six compartments - numbered 1, 4, 5, 6, 7 and 8, that do not fall into the category of "yellow."

The probability is therefore :

= ( Number of not yellow sections ) / ( Total number of sections )

= 6 / 8

= 3 / 4

= 75 %

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The equation of the hyperbola is -(x + 5)²/71 + (y - 2)² = -71

We can also find the distance between the center and the given focus, which is the distance between (-5, 2) and (-5 - √29, 2):

d = |-5 - (-5 - √29)| = √29

Substituting in the known values, we get:

c² = a² + b²

(√29)² = (10)² + b²

29 = 100 + b²

b² = -71

(x - h)²/a² - (y - k)²/b² = 1

where (h, k) is the center of the hyperbola.

Substituting in the known values, we get:

(x + 5)²/100 - (y - 2)²/-71 = 1

Multiplying both sides by -71, we get:

-(x + 5)²/71 + (y - 2)²/1 = -71/1

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Lana paid $36, Pam paid $17.95, and Cheryl paid $20.05, by using substitution or elimination, for the groceries.

Let's start by assigning variables to the unknown quantities in the problem. Let's call the cost of Lana's groceries "L", the cost of Pam's groceries "P", and the cost of Cheryl's groceries "C". We can set up a system of equations based on the information given:

1) P = 0.5L - 0.05 (Pam's groceries were $0.05 cheaper than half of Lana's groceries)

2) C = P + 2.10 (Cheryl's groceries were $2.10 more than Pam's groceries)

3) L + P + C = 74 (the total bill was $74)

We now have three equations with three unknowns, which we can solve using substitution or elimination. Let's use substitution:

Substitute equation 1 into equation 2 for P:

C = (0.5L - 0.05) + 2.10

Simplify:

C = 0.5L + 2.05

Substitute equations 1 and 3 into the equation above:

L + P + C = 74

L + (0.5L - 0.05) + (0.5L + 2.05) = 74

Simplify:

2L + 2 = 74

2L = 72

L = 36

Now that we know the cost of Lana's groceries, we can use equation 1 to find the cost of Pam's groceries:

P = 0.5L - 0.05

P = 0.5(36) - 0.05

P = 17.95

Finally, we can use equation 2 to find the cost of Cheryl's groceries:

C = P + 2.10

C = 17.95 + 2.10

C = 20.05

Therefore, Lana paid $36, Pam paid $17.95, and Cheryl paid $20.05 for the groceries.

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Answer: Pairs (0, 14) and (10, 0

Step-by-step explanation:

Therefore, an initial deposit of $37,728.66 is required to have a college fund of $110,000 in 17 years with quarterly compounding and an APR of 4.9%.

An amount held in an account is referred to as a deposit. It might be put up in a bank as collateral for goods that are being rented out or bought. A deposit is used in many different sorts of economic transactions.

Compound interest can be calculated using the following formula to determine the required down payment:

A = P(1 + r/n)(nt)

where:

A = the future value of the account (in this case, $110,000)

P = the principal or initial deposit

r = the annual interest rate (4.9%)

n = the number of times the interest is compounded per year (4 for quarterly compounding)

t = the number of years (17)

When we enter the specified numbers into the formula, we obtain:

$110,000 = P(1 + 0.049/4)(4*17)

$110,000 = P(1.01225)⁶⁸

$110,000 = P * 2.9126

Dividing both sides by 2.9126, we get:

P = $37,728.66

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For the given vectors u and v, (a) 6u =  (24, 30). (b) 6u+4v =  (40, 30).      (c) v-4u =  (-12, -20).

Given vectors u and v.

a) To find 6u, we simply multiply each component of u by 6:

6u = 6(4, 5) = (6(4), 6(5)) = (24, 30)

Therefore, 6u = (24, 30).

b) To find 6u + 4v, we first need to find 4v by multiplying each component of v by 4:

4v = 4(4, 0) = (4(4), 4(0)) = (16, 0)

Next, we add 6u and 4v by adding the corresponding components:

6u + 4v = (24, 30) + (16, 0) = (24+16, 30+0) = (40, 30)

Therefore, 6u + 4v = (40, 30).

c) To find v - 4u, we first need to find 4u by multiplying each component of u by 4:

4u = 4(4, 5) = (4(4), 4(5)) = (16, 20)

Next, we subtract 4u from v by subtracting the corresponding components:

v - 4u = (4, 0) - (16, 20) = (4-16, 0-20) = (-12, -20)

Therefore, v - 4u = (-12, -20).

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The calculated perimeter of the square from the side length is 17.2x + 8

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

A square that has a side length of 4.3x + 2 inches.

Using the above as a guide, we have the following:

Perimeter = 4 * side length

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

Perimeter = 4 * (4.3x + 2)

When teh brackets are opened, we have

Perimeter = 17.2x + 8

Hence, the perimeter is 17.2x + 8

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If Sarah uses 3/4 yard of ribbon to make a hair bow, she will need 6 and 3/4 yards of ribbon to make 9 hair bows.

To find out how many yards of ribbon Sarah will use to make 9 hair bows, we need to multiply the amount of ribbon used for one hair bow (3/4 yard) by the number of hair bows she wants to make (9).

So, the equation we need to use is:

3/4 yard of ribbon per hair bow x 9 hair bows = ? yards of ribbon

To solve for the answer, we can simplify the equation:

3/4 x 9 = 27/4

So Sarah will need 27/4 yards of ribbon to make 9 hair bows.

To convert this fraction to a mixed number, we can divide the numerator (27) by the denominator (4) and write the remainder as a fraction:

27 ÷ 4 = 6 with a remainder of 3

In summary, Sarah will need 6 and 3/4 yards of ribbon to make 9 hair bows, if she uses 3/4 yard of ribbon to make one hair bow.

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