Solve the system by substitution.
5x + 9y = 21
-6y = x

The main answer to your question is "interchanging". To find the inverse of a function, we interchange the numbers in each ordered pair of the function. This means that we switch the x and y values of each point in the function.

For example, if we have a function f(x) = 2x + 3, the ordered pairs would be (1,5), (2,7), (3,9), etc. To find the inverse function, we would switch the x and y values of each point to get ordered pairs such as (5,1), (7,2), (9,3), etc.

The explanation for why we interchange the numbers is that the inverse function "undoes" the original function. If we apply the original function to a number, the inverse function will take us back to the original number. By switching the x and y values, we make sure that the inverse function will undo the original function.

In conclusion, to find the inverse of a function, we interchange the numbers in each ordered pair of the function. This ensures that the inverse function will undo the original function.
Hi! I'm happy to help you with your question.

Main answer: The inverse of a function can be found by interchanging the numbers in each ordered pair of the function.

Explanation: When finding the inverse of a function, you are essentially swapping the input and output values in each ordered pair (x, y) to create a new ordered pair (y, x). This process is called interchanging the numbers in the ordered pair.

Conclusion: To find the inverse of a function, you need to interchange the numbers in each ordered pair of the function, which essentially swaps the input and output values.

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The relationship between variable F and x is 2x + 12000000/x

Let y represent the length (in feet) of a side perpendicular to the dividing fence, and let x represent the length (in feet) of a side parallel to the dividing fence.

Let F represent the length of fencing in feet.

Area of fencing = 6000000 ft²

Area of fencing = x * (y + y) = x * 2y

Area of fencing = 2xy

6000000=2xy

xy = 3000000

y = 3000000/x

The perimeter is:

F = x + y + y + x + y + y

F = 2x + 4y

F = 2x + 4(3000000/x)

F = 2x + 12000000/x

Therefore the relationship between variable F and x is 2x + 12000000/x

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The value of x in the shape is 111

The shape is a heptagon

This is so because the shape has 7 sides

i.e. n = 7

The sum of angles in the shape is

Sum = 180(n - 2)

So, we have

Sum = 180(7 - 2)

Evaluate

Sum = 900

Also, the sum of angles in the shape is

Sum = x + 120 + 158 + 126 + 125 + 121 + 139

This gives

Sum = x + 789

So, we have

x + 789 = 900

Evaluate the like terms

x = 111

Hence, the value of x in the shape is 111

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The variable P0 represents the initial amount.

In the exponential growth model, P(t) represents the population at time t, P0 represents the initial population at time t=0, k represents the growth rate, and e is the mathematical constant, approximately equal to 2.71828. Therefore, P0 represents the initial amount or the starting point of the population.

The correct answer is B, the variable P0 represents the initial amount.

B. The variable P0 represents the initial amount.

In the exponential growth model P(t) = P0 * e^(kt), the variables are defined as follows:
- P(t): The population size at time 't'
- P0: The initial population size (i.e., the population size at the beginning, when t=0)
- e: The base of the natural logarithm (approximately 2.71828)
- k: The growth rate constant
- t: Time

In this model, P0 is the population size at the beginning of the observed period (when t=0). Therefore, it represents the initial amount.


The variable P0 in the exponential growth model P(t) = P0 * e^(kt) represents the initial amount of the population or quantity being studied.

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Step-by-step explanation:

A parenthesis is used when the number next to it is NOT part of the solution set

   like :   all numbers up to but not including 3 .    

  Parens are always next to  infinity  when it is part of the solution set .

  A bracket is used when the number next to it is included in the solution set.

Answer:

115,200

Step-by-step explanation:

10 powered by 5 is 10000. 100000 times 1.152 is 115,200

Answer:

42,048,000

Step-by-step explanation:

1.152*10^5=115200

multiply that by the number of days in a year (365)

115200*365=42048000

The maintenance cost expected to be incurred during July is $9,400. This corresponds to option C in the given choices.

To determine the maintenance cost expected to be incurred during July, we need to substitute the planned machine time of 9,000 hours into the cost formula Y = $5,800 + $0.40X.

Plugging in X = 9,000 into the formula, we get:

Y = $5,800 + $0.40(9,000)

Y = $5,800 + $3,600

Y = $9,400

Therefore, the maintenance cost expected to be incurred during July is $9,400. This corresponds to option C in the given choices.

The cost formula Y = $5,800 + $0.40X represents the fixed cost component of $5,800 plus the variable cost component of $0.40 per machine-hour. By multiplying the planned machine time (X) by the variable cost rate, we can determine the additional cost incurred based on the number of machine-hours. Adding the fixed cost component gives us the total maintenance cost expected for a given level of machine-time. In this case, with 9,000 hours of planned machine time, the expected maintenance cost is $9,400.

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Step-by-step explanation:

\( { ({3}^{3} )}^{4} \)

\( = {3}^{3 \times 4} \)

\( = {3}^{12} \)

\(or \)

\( = 531441\)

Answer:

Step-by-step explanation:

I think it is helpful to set this up as a proportion as it is the easiest way in my opinion.

1st step: 396/18 and x/11

This is the way you set up the proportion.

2nd Step: You would have to multiply 396 and 11 then divide the value by 18.

396x11 = 4,356

4,356/18 = 242

x=242 Dollars

Answer: 242 dollars

Therefore, Alan would make 242$ while working at the same rate for 11 hours.

The force F = r^2k(r^) is not conservative because its curl is nonzero. The force is central because it depends only on r and acts along the radial direction. Since it is not conservative, there is no potential energy function V(r) associated with this force

To determine whether the force F = r^2k(r^) is conservative and central, let's analyze its properties.

A force is conservative if it satisfies the condition ∇ × F = 0, where ∇ is the gradient operator. In Cartesian coordinates, the force can be written as F = Fx i + Fy j + Fz k, where Fx, Fy, and Fz are the components of the force in the x, y, and z directions, respectively. The curl of F is given by:

∇ × F = (∂Fz/∂y - ∂Fy/∂z)i + (∂Fx/∂z - ∂Fz/∂x)j + (∂Fy/∂x - ∂Fx/∂y)k.

Calculating the components of F = r^2k(r^):

Fx = 0, since there is no force component in the x-direction.

Fy = 0, since there is no force component in the y-direction.

Fz = r^2kr^.

Taking the partial derivatives, we have:

∂Fz/∂x = ∂/∂x (r^2kr^) = 2rkr^2(∂r/∂x) = 2rkr^2(x/r) = 2xkr^3.

∂Fz/∂y = ∂/∂y (r^2kr^) = 2rkr^2(∂r/∂y) = 2rkr^2(y/r) = 2ykr^3.

Substituting these values into the curl equation, we get:

∇ × F = (2ykr^3 - 2xkr^3)k = 2k(r^3y - r^3x).

Since the curl of F is not zero, ∇ × F ≠ 0, we conclude that the force F = r^2k(r^) is not conservative.

Now let's determine if the force is central. A force is central if it depends only on the distance from the origin (r) and acts along the radial direction (r^).

For F = r^2k(r^), the force is indeed central because it depends solely on r (the magnitude of the position vector) and acts along the radial direction r^. Hence, it can be written as F = Fr(r^), where Fr is a function of r.

Since the force is not conservative, it does not possess a potential energy function. In conservative forces, the potential energy function V(r) can be defined, and the force can be expressed as the negative gradient of the potential energy, i.e., F = -∇V. However, since F is not conservative, there is no potential energy function associated with it.

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1) The assumption T(n) = cn² - dn satisfies the recurrence relation. 2) The value of k should be greater than 0. 3) The asymptotic upper bound is O(n^log_4(2)) and O(nlogn). 4) The substitution method fails to provide an asymptotic upper bound for this recurrence.

Question 1:

To show that a substitution proof with the assumption T(n)≤cn² fails for the recurrence T(n) = 4T(n/2) + n, we can substitute T(n) = cn² into the recurrence relation and see if it holds.

T(n) = 4T(n/2) + n

cn² = 4c(n/2)² + n

cn² = 4c(n²/4) + n

cn² = cn² + n

As we can see, the assumption T(n) = cn² does not satisfy the recurrence relation. Therefore, the substitution proof fails.

To make a substitution proof work, we can subtract a lower-order term to modify the assumption. Let's assume T(n) = cn² - dn, where d is a positive constant. Now let's substitute this assumption into the recurrence relation.

T(n) = 4T(n/2) + n

cn² - dn = 4c(n/2)² - 2d(n/2) + n

cn² - dn = cn² - dn + n

This time, the assumption T(n) = cn² - dn satisfies the recurrence relation. By subtracting the lower-order term, we ensure that the assumption matches the recurrence relation and the substitution proof works.

Question 2:

To sketch the recursion tree for the recurrence T(n) = 4T(n/3) + n, we start with the initial value T(n) at the top and recursively expand the tree by dividing n by 3 at each level until we reach the base case.

We continue this process until we reach the base case T(1), where the recursion stops.

Using the substitution method, we assume T(n) = O(\(n^k\)) and substitute it into the recurrence relation.

T(n) = 4T(n/3) + n

O(\(n^k\)) = 4O(\((n/3)^k\)) + n

O(\(n^k\)) = 4O(\(n^k\) * \((1/3)^k\)) + n

By analyzing the equation, we can see that (1/3)^k appears as a constant factor. For the recurrence to be solved with the substitution method, this constant factor must be less than 1. Therefore, the value of k should be greater than 0.

Question 3:

a) T(n) = 2T(n/4) + n

For this recurrence, we can see that a = 2, b = 4, and f(n) = n. Since f(n) = n is a polynomial function and \(log_b\)(a) = \(log_4\)(2) < 1, we can apply case 1 of the master theorem. The asymptotic upper bound is O(\(n^{log_4(2)\)).

b) T(n) = 4T(n/3) + nlogn

In this case, a = 4, b = 3, and f(n) = nlogn. Since f(n) = nlogn is a polynomial multiplied by logn, we can apply case 3 of the master theorem. The asymptotic upper bound is O(nlogn).

Question 4:

The master theorem cannot be directly applied to the recurrence T(n) = 4T(n/2) + n^2logn because the form of the recurrence does not match the standard form required by the master theorem. The master theorem can be applied to recurrences of the form T(n) = aT(n/b) + f(n), where a ≥ 1, b > 1, and f(n) is a polynomial function.

To find an asymptotic upper bound for this recurrence, we can use the substitution method. Let's assume T(n) = O(\(n^k\)) and substitute it into the recurrence relation.

T(n) = 4T(n/2) + \(n^2\)logn

O(\(n^k\)) = 4O(\((n/2)^k\)) + \(n^2\)logn

O(\(n^k\)) = 4O(\(n^k\) * \((1/2)^k\)) + \(n^2\)logn

By analyzing the equation, we can see that \((1/2)^k\) appears as a constant factor. For the recurrence to be solved with the substitution method, this constant factor must be less than 1. However, in this case, the constant factor is 1/2, which is not less than 1. Therefore, the substitution method fails to provide an asymptotic upper bound for this recurrence.

Nevertheless, we can still provide an asymptotic upper bound for the recurrence by using other techniques such as the recursion tree or the iterative method.

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Answer:

Step-by-step explanation:

Step 1: Draw a line from the vertex of the angle to the given point on the x-axis.

Step 2: Draw a line from the given point to the point on the y-axis where the angle intersects the y-axis.

Step 3: Calculate the length of the line segment created in Step 2. This will be x.

Step 4: Calculate the measure of angle 0 by using the given measure of angle and the length of x. This will be 0 degrees.

Yes, the given curves are orthogonal.

In mathematics, an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally.

Given equation is,

x²+y²=ax

Differentiating,

2x+2yy'=a

y' = (a-2x)/2y = m1

Again,

x²+y²=by

Differentiating,

2x+2yy'=by'

y' = -2x/(2y-b) = m2

For both curves are orthogonal, we have

m1*m2 = -1

(a-2x)/2y*-2x/(2y-b)  = -1

-2ax+4x² = -4y²+2yb

4(x²+y²) = 2ax+2yb

Since, ax = (x²+y²)

by = (x²+y²)

Then,

4(x²+y²) = 2(x²+y²) +2(x²+y²)

4(x²+y²) = 4(x²+y²)      (true)

Hence, the above response is appropriate.

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Answer:

x is 7

Step-by-step explanation:

-3-(-8)-(-2) = x

             7 = x

Answer:

slope (m) = 1/5

y-intercept (b) = 3

Step-by-step explanation:

slope (m) = 1/5

y-intercept (b) = 3

The standard matrix of the given linear transformation T is [1 8] [0 1] [0 -1] [1 0].

The question requires us to find the standard matrix of a linear transformation T.

This linear transformation involves two steps: A horizontal shear that transforms e2 into e2 + 8e1 (leaving e1 unchanged) A reflection through the line x2 = -x1

Let's say a vector v in R2 be represented as a column vector (x, y). Now let's apply the given linear transformation T to it. We'll do it in two steps:

Step 1: Applying the horizontal shear to the vector. Recall that T performs a horizontal shear that transforms e2 into e2 + 8e1 (leaving e1 unchanged).

In other words, T(e1) = e1 and T(e2) = e2 + 8e1.

So let's find the image of the vector v under this horizontal shear. Since T is a linear transformation, we can write T(v) as T(v1e1 + v2e2) = v1T(e1) + v2T(e2).

Plugging in the values of T(e1) and T(e2), we get:T(v) = v1e1 + v2(e2 + 8e1) = (v1 + 8v2)e1 + v2e2.

So the image of v under the horizontal shear is given by the vector (v1 + 8v2, v2).

Applying the reflection to the vector. Recall that T also reflects points through the line x2 = -x1.

So if we reflect the image of v obtained in step 1 through this line, we'll get the final image of v under T.

To reflect a vector through the line x2 = -x1, we can first reflect it through the y-axis, then rotate it by 45 degrees, and then reflect it back through the y-axis.

This can be accomplished by the following matrix:  B = [1 0] [0 -1] [0 -1] [1 0] [1 0] [0 -1]

So let's apply this matrix to the image of v obtained in step 1. We have:

(v1 + 8v2, v2)B = (v1 + 8v2, -v2, -v2, v1 + 8v2, v1 + 8v2, -v2)

Multiplying the matrices A and B, we get:A·B = [1 8] [0 1] [0 -1] [1 0]

And this is the standard matrix of T.

Therefore, the standard matrix of the given linear transformation T is [1 8] [0 1] [0 -1] [1 0].

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Answer:

plugging in those values you get 40+(81/9)=40+9=49.

The displacement and distance traveled depend on the given time interval.

To find the displacement, we need to integrate the velocity function from 0 to 5 or 0 to 5π, depending on the given time interval.

If the time interval is [0,5], we have:

Displacement = ∫[0,5] v(t) dt
Displacement = ∫[0,5] 6cos(t) dt
Displacement = [6sin(t)] from 0 to 5
Displacement = 6sin(5) - 6sin(0)
Displacement ≈ 2.785 meters

If the time interval is [0,5π], we have:

Displacement = ∫[0,5π] v(t) dt
Displacement = ∫[0,5π] 6cos(t) dt
Displacement = [6sin(t)] from 0 to 5π
Displacement = 0 - 6sin(0)
Displacement = 0 meters

To find the distance traveled, we need to integrate the absolute value of the velocity function over the given time interval.

If the time interval is [0,5], we have:

Distance = ∫[0,5] |v(t)| dt
Distance = ∫[0,5] |6cos(t)| dt
Distance = ∫[0,π/2] 6cos(t) dt + ∫[π/2,5] -6cos(t) dt
Distance = [6sin(t)] from 0 to π/2 + [-6sin(t)] from π/2 to 5
Distance = 6 - 6sin(5) ≈ 2.215 meters

If the time interval is [0,5π], we have:

Distance = ∫[0,5π] |v(t)| dt
Distance = ∫[0,5π] |6cos(t)| dt
Distance = ∫[0,π] 6cos(t) dt + ∫[π,2π] -6cos(t) dt + ∫[2π,3π] 6cos(t) dt + ∫[3π,4π] -6cos(t) dt + ∫[4π,5π] 6cos(t) dt
Distance = [6sin(t)] from 0 to π + [-6sin(t)] from π to 2π + [6sin(t)] from 2π to 3π + [-6sin(t)] from 3π to 4π + [6sin(t)] from 4π to 5π
Distance = 0 meters

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Answer: No

Step-by-step explanation:

The domain consists of the numbers 6, 5, 6, 2

6 is repeated twice

The criteria for a function is that there are no repeating numbers in the domain

This does not follow the criteria, therefore this is not a function

Answer-No

Answer:

3.75

Step-by-step explanation:

g(x)=60(1-0.5)^x, g(4)

g(4)=60(1-0.5)^4

g(4)=60(.5)^4

g(4)=60(0.0625)

g(4)=3.75

Answer:

the Greatest common factor is 5

Answer:

57

Step-by-step explanation:

they are congruent

Answer:

\(\huge\boxed{57 \textdegree}\)

Step-by-step explanation:

We can use basic angle relationships to find the measure of ∠CDG.

We already know that ∠HGD is 57°. Since Lines CE and FH are parallel, we can tell that ∠HGD and ∠CDG are alternate interior angles. This means that their angle measures will be the exact same.

So, since ∠HGD is 57°, so it ∠CDG.

Hope this helped!

The sampling technique used to obtain the described sample is A. Systematic sampling.

In systematic sampling, the elements of the population are ordered in some way, and then a starting point is randomly selected. From that point, every nth element is selected to be part of the sample.

In the given scenario, the first 35 students leaving the library were selected. This suggests that the students were ordered in some manner, and a systematic approach was used to select every nth student. Therefore, the sampling technique used is systematic sampling.

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Answer:

it is a statistical measure of the relationship between two variables that indicates the extent to which the variables change together in the same or opposite direction. Correlation does not imply causation, meaning that a correlation between two variables does not necessarily mean that one variable causes the other.

Based on this definition, the correct answer is b. Increasing values of X go with either increasing or decreasing values of Y. A correlation exists between two variables when both variables increase or decrease together. This statement captures the idea that correlation can be positive or negative, and that it reflects a linear relationship between two variables.

Step-by-step explanation:

a is wrong because it only describes positive correlation, not negative correlation.

c is wrong because it confuses correlation with consistency. Correlation does not require that higher values of X always go with higher or lower values of Y, only that they tend to do so on average.

d and f are wrong because they assume causation from correlation, which is a logical fallacy.

e is wrong because it contradicts itself. It says that increasing values of X go with increasing values of Y, which is positive correlation, but then it says that a correlation exists when both variables decrease together, which is negative correlation.

If X is correlated with Y, it implies a predictive statistical relationship between X and Y. This correlation can be positive or negative implying respective increase or decrease in values of both variables. But, this correlation doesn't prove causation.

If X is correlated with Y, it indicates a statistical relationship between the two variables, X and Y. This relationship can be positive or negative. If it is a positive correlation, as X increases, Y will also increase and similarly, as X decreases, Y will also decrease. Contrarily, in a negative correlation, as X increases, Y decreases and vice versa. However, it is important to understand that correlation does not imply causation. That is, if X and Y are correlated, it does not necessarily mean that changes in X cause changes in Y or vice versa. It only means that they move in a predictable manner relative to each other.

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The function \($$f(x)= \begin{cases}1-x^2 & x \leqslant 1 \\ \ln (x) & x \geqslant 1\end{cases}$$\)  is continuous on (-∞, ∞).

As per the given data the function f(x) is given by:

\($$f(x)= \begin{cases}1-x^2 & x \leqslant 1 \\ \ln (x) & x \geqslant 1\end{cases}$$\)

Here we have to determine that the function f(x) is continuous on (-∞, ∞)

If we show that f(x) is continuous at x = 1 then f(x) is continuous on (-∞, ∞)

What are continuous function?

A continuous function in mathematics is one where changes in the parameter cause constant changes in the function's value (i.e., a change without a leap). This shows that there are no abrupt changes in value or discontinuities.

To show f(x) is continuous at x = 1

\(\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}}\) f(x)

\(\rightarrow \lim _{x \rightarrow 1^{-}} f(x) & =\lim _{x \rightarrow 1^{-}}\left(1-x^2\right) \\\)

= 1 - 1

= 0

\(\lim _{x \rightarrow 1^{+}} f(x) & =\lim _{x \rightarrow 1^{+}} \ln (x) \\\)

= ln 1

= 0

Therefore  \(\lim _{x \rightarrow 1^{+}} f(x) & =\lim _{x \rightarrow 1^{-}} f(x)-0\).

Hence f(x) is continuous on (-∞, ∞)

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Answer:

I can even see anything can you reupload your answer, please?

Step-by-step explanation:

Thomas Jefferson attempted to fight the spread of slavery by doing what is mentioned in D. "Supporting a law banning slavery in new territories."

Thomas Jefferson was an American Founding Father, politician, and third President of the United States. He was the principal author of the Declaration of Independence and is considered one of the most influential Founding Fathers due to his contributions to the founding of the United States.

Thomas Jefferson did attempt to fight the spread of slavery by supporting a law that would prohibit slavery in new territories. He proposed this in his draft of the Northwest Ordinance in 1784, which would have banned slavery in the Northwest Territory. Although his proposal did not pass, it was eventually adopted in 1787 and helped to establish a precedent for banning slavery in new territories. While Jefferson did oppose the spread of slavery, he did not take more direct action to end the practice, as he himself was a slave owner throughout his life.

With that in mind, the option that seems more plausible is D.

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7x5 -7 x 2 =21

5x-2x =21

Combine like terms

3x=21

Divide both sides by 3

x=21/3

x=7

Answer:

y = (4/3)x

Step-by-step explanation:

You have to find the y-intercept, b. Do this by substituting in the x and y values into the equation:

12 = (4/3)(9) + b

12 = 12 + b

b = 0

So, y = (4/3)x

Hope that helps! :D

Answer: y = 4/3x

y = -3/4x + 1 ⇔ 3x + 4y - 4 = 0

have:

(d): perpendicular to 3x + 4y - 4 = 0

     contains the point (9;12)

=> (d): 4x - 3y + 3.12 - 4.9 = 0 ⇔ 4x - 3y = 0 ⇔ y = 4/3.x

Step-by-step explanation:

By using Python, you will come up with your own challenging algorithm for other students to trace.

If a certain is true, the if/else expression triggers a sequence of instructions to run. Another piece of code may be run if the condition is false.

The if/else statement is a component of Python's "Conditional" Statements, which are used to carry out various operations based on various circumstances.

These conditional statements are available in Python:

If you want a block of code to run only if a certain condition is true, use the if statement.

If the same expression is false, use instead that to provide a set of instructions that should be run.

If the first expression is false, can use else if sentence to establish a comprehensive criterion to test.

To choose which of the several program code should be performed, use the switch.

How to write the code?

num = 100

if num < 20:

print('Less than 20')

if num < 30:

print('Less than 30')

if num < 40:

print('Less than 40')

if num < 50:

print('Less than 50')

if num > 100 or num == 100:

print('More than or equal to 100')

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