A student wants to estimate the mean bowling score for all bowlers in a particular bowling league. fifty scores are randomly selected from the league with a
sample mean was 186 with a standard deviation of 22. assume normality.
5. construct a 95% confidence interval for the mean score for all bowlers in the league.
(179.75, 192.25
(177.66, 194.34)
(180.78, 191.22)
(163.83, 208.17)
(179.9, 192.1)

Answer:

Step-by-step explanation:

(for the first question)

1/2 gallons cover 1/6th of the wall then

1-gallon covers 1/3rd of the wall

so 3 gallons cover one wall

(second question)

you have to calculate 85% of $35 because it is 15% percent off.

35*0.85=29.75

The sale price is $29.75.

(third question)

6*3=18

14.4*100/18

1440/18

80

100-80 = 20

The answer is A (20%)

Answer:

3 gallons of paint

$29.75

20%

Step-by-step explanation:

1. Let's break this down:

1/2 gallon of paint covers 1/6 of his wall.

This means that we have to multiply 1/2 by 6, as there would be 6 1/2 gallon sections of his wall.

1/2·6=3

So, Jamel needs 3 gallons of paint.

2. If a pair of shoes has an original price of $35 but it on sale for 15%, we have to first find how much it's now on sale for:

15/100·35

=0.15·35

=5.25

This is the how much it's off, so subtract 5.25 from 35

35-5.25=$29.75

The shoes have a sale price of $29.75

Even though you said this was wrong, you may have to put a dollar sign in front of it.

3. This is worded a little, but assuming:

1 swimming pass is $6, 3 people buy the swimming pass, and the total cost for the 3 passes in total is $14.40, we have to find out the discount.

So, originally, before the discount, the total amount for the 3 swimming passes would've been $18, but there's been a discount and now they only had to pay $14.40.

To solve, we do the following:

subtract the original price by the sale price

18-14.40=3.6

divide by the original price

3.6/18=0.2

multiply by 100 to get into percent

0.2x100=20%

This means that A is the correct choice.

Hope this helps! :)

The formula for the area of a rectangle is length times width, or in this case (x+6)(x+2) or x^2 + 8x + 12 in standard form.

To simplify this expression, we can use the distributive property to multiply the two binomials:

(x+6)(x+2) = x(x+2) + 6(x+2)

= x² + 2x + 6x + 12

= x² + 8x + 12

To express the area of the entire rectangle, we need to multiply the length (x + 6) by the width (x + 2). This will give us a polynomial in standard form.

Step 1: Write down the expression for the area of the rectangle.
Area = (x + 6)(x + 2)

Step 2: Use the distributive property (also known as the FOIL method) to expand the expression.
Area = x(x + 2) + 6(x + 2)

Step 3: Continue to expand and simplify the expression.
Area = (x² + 2x) + (6x + 12)

Step 4: Combine like terms.
Area = x² + 8x + 12

So the area of the entire rectangle is expressed as the polynomial x² + 8x + 12 in standard form.

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The required answers are  2.5 liters and  1.5 liters per minute.

Part A:

If we know that the pot has a capacity of 10 liters, we can use the equation y = 1.5x + 2.5 to determine how much soup was in the pot to start with, since at x = 0 minutes, no soup has been added yet.

Substituting x = 0 in the equation, we get:

y = 1.5(0) + 2.5

y = 2.5

Therefore, the pot had 2.5 liters of soup to start with.

Part B:

The equation y = 1.5x + 2.5 tells us how much soup is added to the pot in x minutes, so the rate at which the cook fills the pot can be found by taking the derivative of y with respect to x:

dy/dx = 1.5

Therefore, the rate at which the cook fills the pot is 1.5 liters per minute.

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Brandon rode his bike 8.35 miles further than Randy.

To find out how much further Brandon rode his bike last month than Randy, you'll need to subtract Randy's miles from Brandon's miles using the given values.

Step 1: Identify the miles ridden by both Brandon and Randy.
- Brandon rode 56.28 miles.
- Randy rode 47.93 miles.

Step 2: Subtract Randy's miles from Brandon's miles.
- 56.28 miles (Brandon's miles) - 47.93 miles (Randy's miles) = 8.35 miles.

So, last month, Brandon rode his bike more than Randy.

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Answer:

  g(x) = h(x -7) +5

Step-by-step explanation:

Given h(x) defines a parabola that opens upward with a vertex at (-2, -7) and g(x) defines the same parabola with its vertex at (5, -2), you want to express g(x) in terms of h(x).

The graph of f(x) is translated right h units and up k units by ...

  f(x -h) +k

We see that g(x) is a translation of h(x) right by 7 units and up by 5 units. This means (h, k) is (7, 5), and the translated function is ...

  g(x) = h(x -7) +5

__

Additional comment

This is confirmed by the plots in the second attachment.

Answer:  g(x)=h(x-7) +5

Step-by-step explanation:

The graph g(x) has been shifted up 5   (+5)   and right 7

When shift a function, the y change, up/down, goes at end of function

When shift in x direction happens, you take opposite sign so we will do -7

g(x)=h(x-7) +5

The correct statement that compares the value of the 6 in 26.495 and 17.64 is the value of the 6 in 26.495 is 10 times the value of the 6 in 17.64. Therefore, the correct option is D.

This is because the value of a digit is determined by its place in the number. In 26.495, the 6 is in the tenths place, which means it represents 6/10 or 0.6. In 17.64, the 6 is in the hundredths place, which means it represents 6/100 or 0.06. Therefore, the value of the 6 in 26.495 is 0.6 and the value of the 6 in 17.64 is 0.06.

To compare these values, we can divide the value of the 6 in 26.495 by the value of the 6 in 17.64. This gives us 0.6/0.06 = 10. Therefore, the value of the 6 in 26.495 is 10 times greater than the value of the 6 in 17.64 which corresponds to option D.

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The minimum measurement is 44.5 inches and the maximum measurement is 45.5 inches.

To find the minimum and maximum measurements of Joshua's model airplane, we need to subtract and add 0.5 inches to the given length of 45 inches, respectively.

Minimum measurement:

45 - 0.5 = 44.5 inches

Maximum measurement:

45 + 0.5 = 45.5 inches

Therefore, the minimum measurement is 44.5 inches and the maximum measurement is 45.5 inches.

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Answer:p(x)=3x+5

Step-by-step explanation:

f(x)=2x+3

g(x)=x+2

p(x)=(2x+2)+(x+2)= 3x+5

Answer:

p(x)=ax + b

Step-by-step explanation:

Judith's account has less money than Alan's account after 9 months.

We can start by setting up an equation to represent the amount of money in each account after n months, where n is the number of months that have passed:

For Alan's account:

Amount of money after n months = $10,350 - $1,096n

For Judith's account:

Amount of money after n months = $8,400 - $822n

We want to find the value of n for which Judith's account has more money than Alan's account. In other words, we want to find the value of n that satisfies the following inequality:

8,400 - 822n > 10,350 - 1,096n

To solve for n, we can simplify the inequality by combining like terms:

1,096n - 822n > 10,350 - 8,400

274n > 1,950

n > 7.11

Since n represents the number of months, we can round up to the next whole number and conclude that at the end of the 8th month, Judith's account will have more money than Alan's account.

To check this result, we can substitute n=8 into the equations for the amount of money in each account:

For Alan's account: $10,350 - $1,096(8) = $2,942

For Judith's account: $8,400 - $822(8) = $2,256

We can see that Judith's account has less money after 8 months than Alan's account, so the result is not correct.

Let's try again with n=9:

For Alan's account: $10,350 - $1,096(9) = $1,846

For Judith's account: $8,400 - $822(9) = $1,518

We can see that Judith's account has less money than Alan's account after 9 months, so the correct answer is actually that Judith's account never has more money than Alan's account.

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a. The equation is Total Cost = 0.75x + $600

b. Total cost is $60,600 and net profit is $19,400.

c. The breakeven level of sales volume is $2,400.

a. The equation relating total cost and sales volume can be expressed as:

Total Cost = Cost of Goods Sold + Selling Expenses + Fixed Expenses

where Cost of Goods Sold = 60% (100% - 40%) of the sales volume

           Selling Expenses = 15% of the sales volume

           Fixed Expenses = $600.

Therefore, the equation can be written as:

Total Cost = 0.6x + 0.15x + $600
                 = 0.75x + $600

b. For sales of $80,000, we can substitute x = $80,000 into the equation:

Total Cost = 0.75x + $600
                 = 0.75($80,000) + $600
                 = $60,000 + $600
                 = $60,600

To calculate net profit before taxes, we need to subtract the total cost from the sales volume:

Net Profit before Taxes = Sales - Total Cost
                                       = $80,000 - $60,600
                                       = $19,400

c. To find the breakeven level of sales volume, we need to set the net profit before taxes to zero and solve for x:

Net Profit before Taxes = Sales - Total Cost
               0                      = x - (0.6x + 0.15x + $600)
               0                      = 0.25x - $600
            0.25x                  = $600
               x                       = $2,400

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Answer:

Step-by-step explanation:

eeee

Lamar can spend at most $3.01 on a single pair of socks to stay within his budget.

a. The total cost of the socks and shorts can be represented by the expression:

Total cost = Cost of shorts + Cost of 5 pairs of socks

= $19.95 + 5x

where x is the price of one pair of socks.

b. To write an equation that says Lamar spent exactly $35 on the socks and shorts, we can equate the total cost expression to $35:

$19.95 + 5x = $35

To solve for x, we can first subtract $19.95 from both sides:

5x = $15.05

Then, divide both sides by 5:

x = $3.01

So, Lamar spent $19.95 + 5($3.01) = $35 on the socks and shorts.

c. Other possible prices for the socks that would still allow Lamar to stay within his budget of $35 can be found by plugging in values of x that satisfy the inequality:

Cost of 5 pairs of socks = 5x ≤ $15.05

For example, if the socks cost $2.99 per pair, then the total cost would be:

$19.95 + 5($2.99) = $34.90

which is within Lamar's budget.

d. We can write an inequality to represent the amount Lamar can spend on a single pair of socks as:

x ≤ (35 - 19.95)/5

This simplifies to:

x ≤ $3.01

So, Lamar can spend at most $3.01 on a single pair of socks to stay within his budget.

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In a case whereby Zelda has 8 rabbits with which to start an animal farm. If the rabbit population doubles each month, the number of  months that the rabbit population will be 5,800 is 9.5 months.

In order to calculatre the month then we can use the expression y = abⁿ

a = starting number ( 8 rabbits)

b = rate of change = (2)

n = number of months that we need to calculate

y = rabbit population = 5800

The we can substitute to have

5800 = 8 × 2ⁿ

5800 / 8 = 2ⁿ

725 = 2ⁿ

n = 9.5 months.

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The intensity of the Galesburg earthquake was approximately 63.1 times larger than the intensity of the Stony Point earthquake.

To compare the intensities of the Galesburg and Stony Point earthquakes, we can use the Richter scale formula M = log(I/I₀), where M is the magnitude of the earthquake, I is the intensity of the earthquake, and I₀ is a reference intensity.

Given:

Magnitude of the Galesburg earthquake (M₁) = 4.2

Magnitude of the Stony Point earthquake (M₂) = 2.5

To find the intensity ratio between the two earthquakes, we can use the formula:

I₁/I₂ = 10^(M₁ - M₂)

Substituting the given magnitudes into the formula:

I₁/I₂ = 10^(4.2 - 2.5)

Calculating the exponent:

I₁/I₂ = 10^1.7

Using a calculator, we find that 10^1.7 is approximately 50.12.

Therefore, the intensity of the Galesburg earthquake (I₁) was approximately 50.12 times larger than the intensity of the Stony Point earthquake (I₂).

Alternatively, we can also express this as the intensity of the Galesburg earthquake being approximately 63.1 times larger than the intensity of the Stony Point earthquake (since 50.12 is approximately equal to 63.1).

Hence, the intensity of the Galesburg earthquake was approximately 63.1 times larger than the intensity of the Stony Point earthquake.

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The exact area of our rectangle is 2 square units

The total area of the rectangle can be found by adding up the area of each square. We have four unit squares and one half-square, which can be represented as 4 + 1/2. To add these two values, we need to find a common denominator, which is 2.

Thus, we can represent the area of the rectangle as

=> 8/2 + 1/2 = 9/2 square units.

We know that the area of a rectangle is calculated by multiplying its length by its width. Thus, we can represent the area of our rectangle as follows:

A = lw

where A is the area, l is the length, and w is the width.

Now, we need to differentiate this equation with respect to the width w. This means we are finding the rate of change of the area with respect to the width. Using the product rule of differentiation, we get:

dA/dw = l * dw/dw + w * dl/dw

Since the width is constant (it does not change), the second term on the right-hand side of the equation is zero. Thus, we are left with:

dA/dw = l

To calculate the exact area of our rectangle, we can use the concept of limits. We can start by approximating the area of our rectangle with a width of 1 square unit. In this case, the area of the rectangle would be 4 square units.

We can then approach the width of 1/2 of a square unit by taking the limit as the width approaches 1/2. Using the derivative we calculated earlier, we can represent this limit as follows:

lim(w→1/2) A = lim(w→1/2) lw = l * lim(w→1/2) w = 4 * (1/2) = 2 square units

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Answer: 20.054

Step-by-step explanation:

D = 2R

R = (C/2π)

Using the completing the square method, the quadratic equation x²-5x=9 can be simplified to (x-2.5)²=15.25, and the solutions are x=2.5+√15.25 and x=2.5-√15.25.

To solve this quadratic equation using the completing the square method. Here are the steps

Move the constant term (in this case, 9) to the right-hand side of the equation

x² - 5x = 9 becomes x² - 5x - 9 = 0

To complete the square, we need to add and subtract a constant term inside the parentheses. The constant term we add is half of the coefficient of the x-term, squared. In this case, the coefficient of the x-term is -5, so we need to add and subtract (5/2)² = 6.25.

x² - 5x - 9 + 6.25 - 6.25 = 0

Rearrange the terms inside the parentheses to group the perfect square with the x-term

(x² - 5x + 6.25) - 15.25 = 0

Factor the perfect square trinomial inside the parentheses

(x - 2.5)² - 15.25 = 0

Add 15.25 to both sides of the equation

(x - 2.5)² = 15.25

Take the square root of both sides

x - 2.5 = ±√15.25

Add 2.5 to both sides

x = 2.5 ±√15.25

So the solutions to the equation x² - 5x = 9, using the completing the square method, are x = 2.5 + √15.25 and x = 2.5 - √15.25.

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--The given question is incomplete, the complete question is given

"Can anyone help? i need to solve these using the completing the square method

x²-5x = 9"--

The remaining part of the given shape was drawn about to its symmetry. The complete shape obtained is similar to the Hexagon shape.

Given half shape has the following points,

The points which are missing to complete the other symmetry are:

By joining the above missing points, we can obtain the full symmetry of the shape.

To rotate the obtained shape of the Hexagon to 180° about the origin, we have to inverse the above complete symmetry points. It simply means if the above points are having positive values, we can inverse it to negative and vice-versa.

By rotating the obtained shape to 180° about the origin, we can obtain the below following points,

The images are attached below for the complete symmetry shape.

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Given question is not having enough required information, so I am attaching the image of the shape which we have to work on,

The amount the temperature during the day changed between 9 a.m. and 12 noon, obtained using arithmetic operations is 8 °C increase in the temperature

Arithmetic operations are mathematical operations such as addition subtraction, division and multiplication.

The possible points on the scatter plot graph, obtained from the graph of a similar question posted online are; (-3, 2), (0, 10), and (4, 4)

The coordinate point on the graph corresponding to 9 a.m. is (-3, 2)

Therefore, the temperature at 9 a.m. is 2°C

The coordinate point on the graph corresponding to 12 noon is (0, 10)

Therefore, the temperature at 12 noon. is 10°C

The change in temperature between the temperature at 9 a.m. and the temperature at 12 noon = 10°C - 2°C = 8 °C (Increase in temperature)

Please find attached the possible scatter plot in the question

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It would take approximately 7.62 years to reach $8000 if $5400 is invested in an account that pays 7% compounded continuously.

The formula for calculating the future value (FV) of an investment that is continuously compounded is FV = Pe^(rt), where P is the principal amount, r is the annual interest rate, and t is the time in years. In this case, P = $5400, r = 7% = 0.07, and FV = $8000. Substituting these values into the formula, we get:

$8000 = $5400e^(0.07t)

Dividing both sides by $5400 and taking the natural logarithm of both sides, we get:

ln(8000/5400) = 0.07t

Simplifying the left side of the equation, we get:

ln(4/3) = 0.07t

Solving for t, we get:

t = ln(4/3)/0.07 ≈ 7.62 years

Therefore, it would take approximately 7.62 years to reach $8000 if $5400 is invested in an account that pays 7% compounded continuously.

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The value of x is approximately 61.7 degrees using the Law of Sines.

To find the value of x, we can use the Law of

Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides. Mathematically, we can write:

a/sin A = b/sin B = c/sin C

where a, b, and c are the side lengths of the triangle, and A, B, and C are the angles opposite those sides.

Using the given information, we can set up the

equation as follows:

12/sin 42° = 22/sin x

Multiplying both sides by sin x°, we get:

sin x = (22/12) x sin 42°

sin x = 1.6977

Taking the inverse sine of both sides, we get:

x* = sin" (1.6977)

x = 61.7°

Rounding to the nearest tenth, we get:

x = 61.7°

Therefore, the value of x is approximately 61.7

degrees.

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The complete question is:

In the given figure , find the X (round to the nearest tenth) , where the sides are marked as 12cm and 22 cm with the angle of 42°.

The scale would show the weight that is closest to the actual weight of the peaches, whether it is 4.158 or 4.168 pounds.

What is measurement?

Measurement is the process of assigning numerical values to physical quantities such as length, mass, time, temperature, and many others.

It depends on the actual weight of the peaches. If the weight of the peaches is closer to 4.158 pounds, then the scale would show 4.158 pounds. Similarly, if the weight of the peaches is closer to 4.168 pounds, then the scale would show 4.168 pounds.

Since the scale can measure weight to the nearest thousandth of a pound, it can differentiate between weights that differ by one-thousandth of a pound.

Therefore, the scale would show the weight that is closest to the actual weight of the peaches, whether it is 4.158 or 4.168 pounds.

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The statements is true of this categorical proposition is: It is a standard-form categorical proposition because it is a substitution instance of this form: some s are not p option A.

A proposition or statement that is categorically affirmed or denied of all or part of the topic is known as a categorical proposition in syllogistic or classical logic. Hence, there are four fundamental types of categorical propositions: "Every S is P," "No S is P," "Some S is P," and "Some S is not P."

Every man is mortal, for instance, is an A-proposition since these forms are denoted by the letters A, E, I, and O, respectively. In particular, being declarations of reality rather than logical connections, they contrast significantly with hypothetical propositions, such as "If every man is mortal, then Socrates is mortal," which categorical propositions are to be differentiated from and enter into as integral words.

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Limiting space heating would help decrease the greatest carbon-releasing activity in US homes.

From the table, we can see that space heating accounts for the largest percentage of energy use in US homes at 45%. Therefore, by limiting space heating, we can significantly reduce the amount of carbon released into the atmosphere, thus decreasing our carbon footprint.

Other activities like limiting time in hot showers, wearing layers of clothing, and turning off lights and electronics when not in use can also help reduce our carbon footprint, but they are not as effective as limiting space heating.

Additionally, we can consider using energy-efficient heating systems, improving insulation, and reducing air leaks to further reduce our energy use and carbon footprint.

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So, depending on the original coordinates, moving down 9 units and left 2 units will end up in either the third or fourth quadrant of the coordinate plane.

A coordinate is a set of values that specifies the position of a point or an object in a geometric space. In a two-dimensional space, such as the Cartesian plane, a coordinate is typically represented by a pair of numbers (x, y), where x represents the horizontal position (or abscissa) of the point and y represents the vertical position (or ordinate) of the point.

Here,

Starting from an arbitrary point (x, y), if we move down 9 units and left 2 units, we will end up at the point with coordinates (x - 2, y - 9). The new x-coordinate is obtained by subtracting 2 from the original x-coordinate, since we moved 2 units to the left. The new y-coordinate is obtained by subtracting 9 from the original y-coordinate, since we moved 9 units down.

The quadrant we end up in depends on the original coordinates (x, y) and the direction of the movement. If we start in the first quadrant (x > 0, y > 0) and move down and left, we will end up in the third quadrant (x < 0, y < 0).

If we start in the second quadrant (x < 0, y > 0) and move down and left, we will also end up in the third quadrant (x < 0, y < 0).

If we start in the third quadrant (x < 0, y < 0) and move down and left, we will end up in the fourth quadrant (x > 0, y < 0).

If we start in the fourth quadrant (x > 0, y < 0) and move down and left, we will still end up in the fourth quadrant (x > 0, y < 0).

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If Susan stands 300 m away from the lighthouse and looks up at it with an angle of elevation of 5 degrees, the height of the lighthouse is approximately 26 meters. This calculation is important for navigation and other purposes.

To determine the height of the lighthouse, we can use trigonometry. We know that the angle of elevation, which is the angle between Susan's line of sight and the ground, is 5'. We also know the distance between Susan and the lighthouse, which is 300 m.

We can use the tangent function to find the height of the lighthouse:

tan(5') = height/300

To solve for height, we can cross-multiply and simplify:

height = 300 x tan(5')

Using a calculator, we get that the height of the lighthouse is approximately 26.2 m.

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The probability of getting a prime number on the spinner is 2/4 or 1/2.

There are four possible outcomes on the spinner: 6, 7, 8, and 9. To determine the probability of getting a prime number, we need to first identify which of these numbers are prime. Prime numbers are numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder.

Out of the four possible outcomes, only two of them are prime: 7 and 9. Therefore, the probability of getting a prime number is the number of favorable outcomes (2) divided by the total number of possible outcomes (4), which simplifies to 1/2.

To see why this is true, we can think of the probability as a fraction where the numerator is the number of ways to get a prime number and the denominator is the total number of possible outcomes. In this case, there are two ways to get a prime number (7 and 9), and four possible outcomes (6, 7, 8, and 9). Therefore, the probability r is 2/4 or 1/2.

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The number of 3-teams is different, and there is at least one 2-team and one 3-team, fulfilling the requirements.

a) To list all different 3-teams that the friends (A, B, C, D, E) could enter, we can find all the possible combinations of choosing 3 friends out of 5. These combinations are:

1. ABC
2. ABD
3. ABE
4. ACD
5. ACE
6. ADE
7. BCD
8. BCE
9. BDE
10. CDE

b) To maximize the number of teams with exactly two 3-teams and at least one 2-team, we can form the following teams:

1. ABC (3-team)
2. ADE (3-team)
3. BC (2-team)

Here, we have formed 1 two-team and 2 three-teams.

c) To maximize the number of teams with exactly three 3-teams and at least one 2-team, we can form the following teams:

1. ABC (3-team)
2. ADE (3-team)
3. BCE (3-team)
4. CD (2-team)

Here, we have formed 1 two-team and 3 three-teams.

d) The friends want to enter 8 teams, including at least one 2-team and at least one 3-team. We can find three ways of doing this with a different number of 3-teams in each case:

1. Two 3-teams: ABC, ADE (3-teams); BC, BD, BE, CD, CE, DE (2-teams)
2. Three 3-teams: ABC, ADE, BCE (3-teams); AC, AD, AE, BD, BE, CD (2-teams)
3. Four 3-teams: ABC, ADE, BCE, BCD (3-teams); AB, AC, AD, AE (2-teams)

In each case, the number of 3-teams is different, and there is at least one 2-team and one 3-team, fulfilling the requirements.

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The answers to the questions on linear combination have been solved below

1. We can start by multiplying both sides by -2 to eliminate the x-term:

2x + 4y = 0

-2(2x + 4y) = -2(0)

-4x - 8y = 0

Now, we have:

-4x - 8y = 0

9x + 4y = 28

We can now use linear combination by adding these two equations to eliminate the y-term:

(-4x - 8y) + (9x + 4y) = 0 + 28

5x = 28

Dividing both sides by 5, we get:

x = 28/5

Now, we can substitute this value of x into either of the original equations to solve for y. Let's use the second equation:

2x + 4y = 0

2(28/5) + 4y = 0

56/5 + 4y = 0

4y = -56/5

y = -14/5

Therefore, the solution to the system of equations is:

x = 28/5

y = -14/5

We can check that these values satisfy both equations:

9x4y = 28

9(28/5)(-14/5) = 28

-352/25 = 28/25 (true)

2x + 4y = 0

2(28/5) + 4(-14/5) = 0

56/5 - 56/5 = 0 (true)

Therefore, the solution is verified.

2. The system of equations is:

5x + 3y = 41

3x - 6y = 9

We can simplify the second equation by dividing both sides by 3:

3x - 6y = 9

x - 2y = 3

Now we can use linear combination by multiplying the first equation by 2 to eliminate the y-term:

2(5x + 3y) = 2(41)

10x + 6y = 82

(x - 2y) + (10x + 6y) = 3 + 82

11x = 85

Dividing both sides by 11, we get:

x = 85/11

Now we can substitute this value of x into either of the original equations to solve for y. Let's use the first equation:

5x + 3y = 41

5(85/11) + 3y = 41

425/11 + 3y = 41

3y = 41 - 425/11

3y = (451 - 425)/11

y = 26/33

Therefore, the solution to the system of equations is:

x = 85/11

y = 26/33

We can check that these values satisfy both equations:

5x + 3y = 41

5(85/11) + 3(26/33) = 41

425/11 + 26/11 = 41

451/11 = 41 (true)

3x - 6y = 9

3(85/11) - 6(26/33) = 9

255/11 - 52/11 = 9

203/11 = 9 (true)

Therefore, the solution is verified.

3.

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

3x - 6y = -24

3x - 6y = -12

3x - 6y = -24

Subtracting the second equation from the first equation, we get:

0 = 12

This is a contradiction, since 0 cannot equal 12. Therefore, there is no solution that satisfies both equations.

This means that the system is inconsistent, and there are no solutions.

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