L.15 Solve matrix equations using inverses 200 If M=[[7,5,-5]] -8 and N=[[0,7,-7]] 2 xM=N

The number which is a cube number, an odd number, middle digit is prime and is of less than 6 digit is 1331.

The only number that satisfies all the given conditions is 1331.

The number 1331 is a cube number because it is 11 to the power of 3,

which is written as ⇒ (11 × 11 × 11 = 1331).

The number 1331 is also a square number because it is 11 to the power of 2, which means

⇒ (11 × 11 = 121), and 121 is a square number.

The number 1331 is not divisible by 2, so it is an odd number.

The middle digit of the number 1331 is 3, which is a prime number.

And also, the number 1331 has less than 6 digits.

Therefore, the required number is 1331.

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The given question is incomplete, the complete question is

I am a cube number, a square number, an odd number ,my middle digit is prime and i have less than 6 digits, Find that number.

As a result, there is a 6% sales tax.

Percentages is a relative figure used to represent hundredths of a quantity. Because one percent (symbolised as 1%) represents one hundredth of something, 100 percent stands for everything, and 200 percent refers to twice the amount specified. percentage. Percentile is a related topic in mathematics.

The difference in sales tax is the price of the snowboarder with tax versus the price of the snowboarder without tax.

Sales tax therefore equals $848 - $800 = $48.

We need to multiply the result by 100 to get the sales percentage of tax, which we can then divide by the price of the snowboard before taxes.

Sales tax percentage = (Sales tax / Cost without tax) x 100

= ($48 / $800) x 100

= 0.06 x 100

= 6%

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To calculate the effect of the pump, use the formula: P = Q * ΔP / η and To calculate the efficiency of the pump, use the formula: η = P / P_e. Use the data provided in the EXCEL file.

To solve the questions, follow these steps:

a) Calculate effects and efficiencies for all operating points:

1. To calculate the effect of the pump, use the formula: P = Q * ΔP / η, where P is the power of the pump, Q is the volume flow rate, ΔP is the pressure difference across the pump, and η is the efficiency of the pump.

2. To calculate the efficiency of the pump, use the formula: η = P / P_e, where P is the power of the pump, and P_e is the electrical power input to the pump.

3. Use the data provided in the EXCEL file to calculate the effects and efficiencies for all operating points.

b) Create a graph showing the delivery pressure (Pa) ("Pump 1") as a function of volume flow (Flow Rate Q) in (m3 / s):

1. Select the data for the delivery pressure (Pa) and volume flow (Flow Rate Q) in the EXCEL file.

2. Click on the "Insert" tab in EXCEL and select the "Scatter" chart option.

3. Choose the "Scatter with Straight Lines" option to create a graph showing the delivery pressure (Pa) as a function of volume flow (Flow Rate Q).

c) Create a graph showing motor power (Pump 1) (W) as a function of volume flow (Flow Rate) Q (l / s) on an axis and efficiency (Pump 1 (%) on a secondary axis:

1. Select the data for the motor power (Pump 1) (W) and volume flow (Flow Rate) Q (l / s) in the EXCEL file.

2. Click on the "Insert" tab in EXCEL and select the "Scatter" chart option.

3. Choose the "Scatter with Straight Lines" option to create a graph showing the motor power (Pump 1) (W) as a function of volume flow (Flow Rate) Q (l / s).

4. Add a secondary axis to the graph by clicking on the "Format" tab in EXCEL and selecting the "Secondary Axis" option.

5. Select the data for the efficiency (Pump 1 (%) and add it to the secondary axis of the graph.

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The total fluid intake for the meal is 493.055 mL

Converting:

1/3 glass of orange juice

= (1/3) x 8 oz = 2.67 oz

= 79.027 mL (rounded to 3 decimal places)

1/2 cup of tea

= (1/2) x 6 oz

= 3 oz = 88.720 mL (rounded to 3 decimal places)

1/2 pint of milk

= (1/2) x 16 oz

= 8 oz = 236.588 mL (rounded to 3 decimal places)

1 tuna fish sandwich = no fluid intake

1 popsicle (3oz)

= 3 oz

= 88.720 mL (rounded to 3 decimal places)

Total fluid intake

= 79.027 mL + 88.720 mL + 236.588 mL + 0 mL + 88.720 mL

= 493.055 mL (rounded to 3 decimal places)

Therefore, the total fluid intake for the meal is 493.055 mL.

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The algebraic equation that could be used to solve the problem is C. n + 4n = 2.10.

An algebraic equation is a mathematical statement that represents the equality of two expressions, using one or more variables, and mathematical operations such as addition, subtraction, multiplication, and division.

To solve this problem, we can use algebraic equations. Let's use the variable "n" to represent the number of nickels that Peggy has.

From the problem statement, we know that Peggy has four times as many quarters as nickels, which means she has 4n quarters.

We also know that she has a total of $2.10 in all. The value of her nickels is 0.05n dollars each, and the value of her quarters is 0.25(4n) = n dollars each. So we can set up the following equation:

0.05n + 0.25(4n) = 2.10

Simplifying this equation, we get:

0.05n + n = 0.525

1.05n = 2.10

n = 2

Therefore, Peggy has 2 nickels and 4(2) = 8 quarters.

Hence, the equation that could be used to solve the problem is C. n + 4n = 2.10.

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The value of the painting at t = 4 is $400 (V(4) = $400) and the value of the painting at t = 6 is $800 (V(6) = $800).

To find the value of the painting at t = 4 and t = 6, we can simply plug in these values into the function V(t) = 100 • 2^t/2, for t >= 0 where t is the number of years since the painting was purchased, and V(t) is its value (in dollars) at time t, and simplify.

For t = 4, we have:
V(4) = 100 • 2^4/2
V(4) = 100 • 2^2
V(4) = 100 • 4
V(4) = 400

For t = 6, we have:
V(6) = 100 • 2^6/2
V(6) = 100 • 2^3
V(6) = 100 • 8
V(6) = 800

Therefore, V(4) = $400 and V(6) = $800.

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The maximum number of sports drinks that can be purchased for $80 is 90 when they are priced at $0.70 per drink. (option B)

Let's start by finding the number of sports drinks that can be purchased for $80 when they are priced at $0.80 per drink.

The price of each drink is $0.80, and the shipping and handling cost is $10.

Let x be the number of sports drinks that can be purchased for $80:

0.80x + 10 = 80

Subtracting 10 from both sides:

0.80x = 70

Dividing both sides by 0.80:

x = 87.5

Since we cannot purchase a fraction of a sports drink, we must round down to the nearest whole number, which is 87.

Now let's check if we can purchase more sports drinks for $80 when they are priced at $0.70 per drink.

The price of each drink is $0.70, and the shipping and handling cost is $17.

Let y be the number of sports drinks that can be purchased for $80:

0.70y + 17 = 80

Subtracting 17 from both sides:

0.70y = 63

Dividing both sides by 0.70:

y = 90

Since we cannot purchase a fraction of a sports drink, we must round down to the nearest whole number, which is 90.

Therefore, the maximum number of sports drinks that can be purchased for $80 is 90 when they are priced at $0.70 per drink.

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a. The probability that brand A will be preferred as the best is 0.65 or 65%.

b. The probability that brand A will be preferred as the best and brand B as the second best is 0.22 or 22%.

c. The probability that brand B will be preferred as the second best given brand A is preferred as the best is 0.615.

What is probability?

Calculating the likelihood of experiments happening is one of the branches of mathematics known as probability. We can determine everything from the likelihood of receiving heads or tails when tossing a coin to the likelihood of making a research blunder, for instance, using a probability.

a) The probability that brand A will be preferred as the best can be calculated by adding up the frequencies of outcomes where brand A is listed first, which are (A, B, C), (A, C, B), and (C, A, B), and dividing by the total number of outcomes:

P(A is preferred as the best) = (220 + 180 + 250) / 1000 = 0.65

Therefore, the probability that brand A will be preferred as the best is 0.65 or 65%.

b) The probability that brand A will be preferred as the best and brand B as the second best can be calculated by adding up the frequency of the outcome (A, B, C), where brand A is listed first and brand B is listed second, and dividing by the total number of outcomes:

P(A is preferred as the best and B is preferred as the second best) = 220 / 1000 = 0.22

Therefore, the probability that brand A will be preferred as the best and brand B as the second best is 0.22 or 22%.

c) To find the probability that brand B will be preferred as the second best given brand A is preferred as the best, we need to consider only the outcomes where A is listed first:

P(B second | A first) = (A, B, C) + (A, C, B) / P(A first)

P(B second | A first) = (220 + 180) / 650

So the probability that brand B will be preferred as the second best given brand A is preferred as the best is 400/650 = 0.615.

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The factored form of the given polynomial 9x^(2)y+9x^(2)-9x^(3)y is 9x^(2)(y+1-xy).

We can factor out the greatest common factor from this polynomial by finding the highest power of each variable that is common to all terms.

The greatest common factor of 9x^(2)y, 9x^(2), and 9x^(3)y is 9x^(2). Therefore, we can factor out 9x^(2) from the given polynomial.

[tex]9x^{2}y+9x^{2}-9x^{3}y = 9x^{2}(y+1-xy)[/tex]

Therefore, the answer is 9x^(2)(y+1-xy).

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The inequality of the scenario is 8.50n + 9 ≤ 43

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

Amount = $43

The cost of n books and 4 pens can be expressed as:

8.50n + 2.25(4)

We can simplify this expression:

8.50n + 9

We know the student has a total of $43 to spend, so we can set up an inequality:

8.50n + 9 ≤ 43

Subtracting 9 from both sides, we get:

8.50n ≤ 34

Dividing both sides by 8.50, we get:

n ≤ 4

Hence, the inequality that best represents the scenario is: n ≤ 4 or 8.50n + 9 ≤ 43

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Complete question

For this item, select the answers from the drop-down menus A student has $43 in total to buy books that cost \$8.50 each and pens that cost $2.25 each. The student buys n number of books and 4 pens

Complete the inequality to best represent the scenario.

The main of the given function f(x) is all real numbers except x=-4.

To find the main, we need to find the roots of the function. That is, we need to find out the values of x for which the value of f(x) is equal to 0.


Solving for the roots, we get the following equation:
8x2-9x+1=0


Solving this equation using the quadratic formula yields:
x = (-9 ± √73)/16


Therefore, the roots of the equation are:
x = 0.41 and -4.41


Since the only root which belongs to all real numbers is x = -4, it is the main of the given function.

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Based on the ratio of red to white paint in making pink paint of 1:3, the total cost to Robert to buy enough paint to make 12 liters is $54.

The Ratio refers to the relative size of one quantity compared to another.

Ratios are expressed in standard ratio form using (:) with one value in relation to another.

Ratios can also be depicted in percentages, fractions, and decimals.

The Ratio of mixing red and white paint = 1:3

The sum of ratios = 4 (1 + 3)

To mix 12 liters of pink paint, the quantity of red paint = 3 liters (12 x 1/4)

To mxi 12 liters of pink paint, the quantity of white paint required = 9 liters (12 x 3/4)

Cost of red paint per 500ml = £5

The cost of 1 liter of red paint = £10 (£5 x 2)

The total cost of 3 liters of red paint = £30 (£10 x 3)

Cost of white paint per 3 liters = £8

The total cost of 9 liters of white paint = £24 (£8 x 9/3)

The total cost of 12 liters of pink paint mixture = £54 (£30 + £24)

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Therefore, the volume of the larger cylinder is 44.8. [tex]m^{3}[/tex], which is closest to option (c) 437.5. [tex]m^{3}[/tex].

Volume is a measure of the amount of space occupied by an object or substance. It is a three-dimensional quantity that is usually measured in cubic units, such as cubic meters (m³), cubic centimeters (cm³), cubic inches (in³), or gallons (gal).

Given by the question.

The ratio of the volumes of two similar solids is the cube of the ratio of their corresponding side lengths. In this case, the ratio of the scale factor of the cylinders is 5:2, so the ratio of their volumes is (5/2) ^3 = 125/8.

If the volume of the smaller cylinder is 28 m^3, then we can set up the following equation to solve for the volume of the larger cylinder:

(125/8) * V = 28

where V is the volume of the larger cylinder.

Simplifying this equation, we get:

V = 28 * (8/125) * 125

V = 224/5

V = 44.8 [tex]m^{3}[/tex]

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25344 ft

We can use the conversion factors to convert 8 kilometers to another measure.

From the given conversion factors, we know that:

1 km ≈ 0.6 mi

1 mi = 5,280 ft

So, to convert 8 kilometers to feet, we can first convert it to miles and then convert miles to feet:

8 km x 0.6 mi/km = 4.8 miles

4.8 miles x 5,280 ft/mile = 25,344 ft

Therefore, 8 kilometers is equivalent to 25,344 feet.

The answer is option (C) 25,344 ft

The solution for \( X \) is \( X = (A - D C)^{-1} D B \).

Assume \( A, B, C, D \), and \( X \) are \( n \times n \) matrices, and assume invertibility of matrices wherever needed. Then, the equation \( A X(B+C X)^{-1}=D \) can be solved for \( X \) by following these steps:

Step 1: Multiply both sides of the equation by \( (B+C X) \) to get rid of the inverse:
\( A X = D(B+C X) \)

Step 2: Distribute the \( D \) on the right side of the equation:
\( A X = D B + D C X \)

Step 3: Rearrange the equation to isolate \( X \) on one side:
\( A X - D C X = D B \)

Step 4: Factor out \( X \) on the left side of the equation:
\( (A - D C) X = D B \)

Step 5: Multiply both sides of the equation by the inverse of \( (A - D C) \) to solve for \( X \):
\( X = (A - D C)^{-1} D B \)

Therefore, the solution for \( X \) is \( X = (A - D C)^{-1} D B \).

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Answer:

To compare Arthur's long jump to the record, we need to convert both measurements to the same unit. Let's convert both measurements to inches:

Record: 22 ft 6 3/4 in = (22 x 12) + 6 + 3/4 = 270 + 6 + 0.75 = 276.75 in

Arthur's jump: 22 ft 5 1/2 in = (22 x 12) + 5 + 1/2 = 270 + 5 + 0.5 = 275.5 in

To determine how much farther Arthur needs to jump to break the record, we subtract Arthur's jump distance from the record distance:

Record distance - Arthur's jump distance = 276.75 in - 275.5 in = 1.25 in

However, we are told that long jumps are measured to the nearest quarter inch. Therefore, we need to round the difference to the nearest quarter inch. Since 1.25 inches is closer to 1.25 than it is to 1.5, we round down to the nearest quarter inch. This gives us:

1.25 in ≈ 1.25/4 = 0.3125 quarters ≈ 0.25 quarters

Therefore, Arthur needs to jump an additional 0.25 quarters (or 1/16 of an inch) to break the record.

Answer:

Step-by-step explanation:

where is the image don't see

The dependent variable and independent variable in the 15th question is

t = dependent and m = independent variable.

A variable is a quantity that may change within the context of a mathematical problem or experiment.

Given is a statement, The temperature, t, of water and the number of minutes, m, the water is in the freezer, we need to identity the dependent variable and independent variable for this statement,

Independent variable :-

It is a variable that doesn't change by the other variables.

Dependent variable :-

The dependent variable is the variable that is used for tested in an experiment.

Here, the temperature of the water is dependent on the number of minutes it has been kept in the freezer,

So, the temperature of the water is the dependent variable and the number of minutes it has been kept in the freezer is the independent variable.

Hence, the dependent variable and independent variable in the 15th question is t = dependent and m = independent variable.

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The question 17 is not completely mentioned

To find (f ∘ g)(x), we need to substitute g(x) into f(x) wherever we see x in the expression for f(x).

So we have:

f(g(x)) = f(3x + 1) = (3x + 1)^2 - 8 = 9x^2 + 6x - 7

Therefore, the correct answer is: 9x^2 + 6x - 7.

Part A: The height of the container is 5cm.

Part B: The cost of the coffee is $2.83.

Part C: The height of the container is 9cm.

Part D: The cost of the hot chocolate powder is $49.35.

What is volume of cylinder ?

The volume of a cylinder is the amount of space occupied by a cylindrical shape. It is given by the formula:

V = πr²h

where V is the volume, r is the radius of the circular base of the cylinder, and h is the height of the cylinder. The formula is derived by multiplying the area of the circular base (πr²) by the height (h) of the cylinder.

According to the question:
Part A:

Given that the container is a cylinder with a radius of 3cm, we can use the formula for the volume of a cylinder to find the height of the container. The formula for the volume of a cylinder is:

V = πr²h

where V is the volume of the cylinder, r is the radius of the cylinder, and h is the height of the cylinder.

Substituting the given values, we get:

45π = π(3)²h

Simplifying and solving for h, we get:

h = 5

Therefore, the height of the container is 5cm.

Part B:

The volume of the container is 45π cm³ and the cost of coffee is $0.02 per cubic centimeter. Therefore, the total cost of the coffee is:

Total cost = Volume x Cost per unit volume

Total cost = 45π x $0.02

Total cost = $0.90π

Rounding to two decimal places, we get:

Total cost = $2.83

Therefore, the cost of the coffee is $2.83.

Part C:

Given that the container is a cylinder with a radius of 5cm, we can use the same formula for the volume of a cylinder to find the height of the container. Substituting the given values, we get:

225π = π(5)²h

Simplifying and solving for h, we get:

h = 9

Therefore, the height of the container is 9cm.

Part D:

The volume of the container is 225π cm³ and the cost of hot chocolate powder is $0.07 per cubic centimeter. Therefore, the total cost of the hot chocolate powder is:

Total cost = Volume x Cost per unit volume

Total cost = 225π x $0.07

Total cost = $15.75π

Rounding to two decimal places, we get:

Total cost = $49.35

Therefore, the cost of the hot chocolate powder is $49.35.

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The correct answer is D. 3x + 2 < -7 or 3x + 2 > 7.

The correct compound inequality that can be used to solve the inequality |3x+2|>7 is D. 3x + 2 < -7 or 3x + 2 > 7.


When solving an absolute value inequality, we need to remember that the absolute value of a number is always positive. This means that if |3x+2|>7, then 3x+2 must be either greater than 7 or less than -7.

To write this as a compound inequality, we can use the word "or" to indicate that either one of these conditions must be true. This gives us the compound inequality 3x + 2 > -7 or 3x + 2 > 7.

However, we need to be careful with the first part of the compound inequality. Since we know that 3x+2 must be less than -7, we need to use the less than symbol (<) instead of the greater than symbol (>). This gives us the correct compound inequality, 3x + 2 < -7 or 3x + 2 > 7.

Therefore, the correct answer is D. 3x + 2 < -7 or 3x + 2 > 7.

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The answer is yes, more than one triangle can be formed using the three angle measures of triangle QPC. This is because the sum of the three angle measures of a triangle is always equal to 180 degrees.

A triangle is a three-sided geometric figure, consisting of three straight lines connecting three vertices. It is one of the most fundamental shapes in geometry and is used as the basis for a variety of mathematical concepts.

If two angle measures are given, any third angle measure between 0 and 180 degrees can be chosen to form a triangle. Thus, if two angle measures of triangle QPC are given, any third angle measure between 0 and 180 degrees can be used to form a triangle.

For example, if two angle measures of triangle QPC are 45 degrees and 65 degrees, then any third angle measure between 0 and 180 degrees can be chosen to form a triangle. Thus, if the third angle measure chosen is 70 degrees, then a triangle with three angle measures of 45 degrees, 65 degrees and 70 degrees can be formed. Similarly, if the third angle measure chosen is 30 degrees, then a triangle with three angle measures of 45 degrees, 65 degrees and 30 degrees can be formed.

Thus, it can be concluded that more than one triangle can be formed using the three angle measures of triangle QPC.

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The root(s) and multiplicity for each term are:
- Root: x = -3/2, Multiplicity: 4
- Root: x = 10, Multiplicity: 2
- Root: x = -1/2, Multiplicity: 6

The root(s) and multiplicity for each term are as follows:

- The term (2x+3) has a root of x = -3/2 and a multiplicity of 4. This means that the value of x = -3/2 makes the term (2x+3) equal to zero and that this root appears 4 times in the equation.
- The term (x-10) has a root of x = 10 and a multiplicity of 2. This means that the value of x = 10 makes the term (x-10) equal to zero and that this root appears 2 times in the equation.
- The term (2x+1) has a root of x = -1/2 and a multiplicity of 6. This means that the value of x = -1/2 makes the term (2x+1) equal to zero and that this root appears 6 times in the equation.

In summary, the root(s) and multiplicity for each term are:
- Root: x = -3/2, Multiplicity: 4
- Root: x = 10, Multiplicity: 2
- Root: x = -1/2, Multiplicity: 6

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Answer:Latasha should ask her mom for help

Step-by-step explanation:

The first four terms of the sequence generated by g_(1)=(1)/(2) g_(n)=6g_(n-1), for all integers n>=2 are (1/2), 3, 18, and 108.

The first four terms of the sequence generated by g_(1)=(1)/(2) g_(n)=6g_(n-1), for all integers n>=2 are (1/2), 3, 18, and 108.

To find the first four terms of the sequence, we can use the given formula and plug in the values for n.

For the first term, n=1, so we use the formula g_(1)=(1)/(2) to get the first term, which is (1/2).

For the second term, n=2, so we use the formula g_(n)=6g_(n-1) and plug in the value for n and the value of the first term for g_(n-1):

g_(2)=6g_(2-1)=6g_(1)=6(1/2)=3

For the third term, n=3, so we use the same formula and plug in the value for n and the value of the second term for g_(n-1):

g_(3)=6g_(3-1)=6g_(2)=6(3)=18

For the fourth term, n=4, so we use the same formula and plug in the value for n and the value of the third term for g_(n-1):

g_(4)=6g_(4-1)=6g_(3)=6(18)=108

Therefore, the first four terms of the sequence are (1/2), 3, 18, and 108.

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Malcom ate 7/10 of the total chocolates.

Amount is a term used to describe the quantity of something, usually a numerical value. It can refer to money, a number of people or objects, or a measurement such as weight or volume. Amounts can also be used to represent a value of something abstract, such as time, effort, or emotion. Amounts are typically quantified, meaning they can be easily compared to other amounts.

The fraction of chocolates that Malcom ate is 5/7 of the milk chocolates and 2/5 of the dark chocolates, so the total fraction of chocolates that Malcom ate is (5/7) + (2/5) = 7/10. Therefore, Malcom ate 7/10 of the total chocolates.

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We have that, given the equation (x)/(7)+(x)/(9)=(9)/(14) we will get as an answer x = 567/224

To solve the equation (x)/(7)+(x)/(9)=(9)/(14), we need to find the common denominator of the fractions on the left-hand side of the equation. The common denominator of 7 and 9 is 63, so we can multiply each fraction by the appropriate factor to get the common denominator:

[tex](x)/(7) * (9)/(9) + (x)/(9) * (7)/(7) = (9)/(14)[/tex]

[tex](9x)/(63) + (7x)/(63) = (9)/(14)[/tex]

Now we can combine the numerators on the left-hand side of the equation:

[tex](16x)/(63) = (9)/(14)[/tex]

To isolate the variable x, we can cross-multiply and then divide by the coefficient of x:

[tex](16x)(14) = (9)(63)[/tex]

[tex]224x = 567[/tex]

[tex]x = 567/224[/tex]

To check our answer, we can plug it back into the original equation and simplify:

[tex](567/224)/(7) + (567/224)/(9) = (9)/(14)[/tex]
[tex](567/224) * (1/7) + (567/224) * (1/9) = (9)/(14)(567/1568) + (567/2016) = (9)/(14)(1275752 + 1140480)/(3167232) = (9)/(14)(2416232)/(3167232) = (9)/(14)(9)/(14) = (9)/(14)[/tex]

Since both sides of the equation are equal, our answer is correct.

Answer: [tex]x = 567/224[/tex]

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Answer:

10

Step-by-step explanation:

To find out how many students participate in both sports and music, we can use the formula:

Total = Group A + Group B - Both

where "Total" is the total number of students participating in sports or music, "Group A" is the number of students participating in sports, "Group B" is the number of students participating in music, and "Both" is the number of students participating in both.

Plugging in the given values, we get:

Total = 100

Group A = 60

Group B = 50

Both = ?

100 = 60 + 50 - Both

Simplifying the equation, we get:

Both = 60 + 50 - 100

Both = 10

Therefore, 10 students participate in both sports and music.

Let V be a finite-dimensional vector space and let S1 and S2 be two bases of V. To show that S1 and S2 have the same number of elements, we will assume, without loss of generality, that S1 has more elements than S2, i.e., n > m. We will then derive a contradiction.

Since S1 is a basis of V, every vector in V can be expressed as a linear combination of the vectors in S1. In particular, for each j = 1, 2, ..., m, we can express βj as a linear combination of the vectors in S1:

βj = c1,jα1 + c2,jα2 + ... + cn,jαn

where c1,j, c2,j, ..., cn,j are scalars. We can write this in matrix form as

| β1 | | c1,1 c1,2 ... c1,m | | α1 |

| β2 | | c2,1 c2,2 ... c2,m | | α2 |

| ... | = | ... ... ... ... | * | ... |

| βm | | cm,1 cm,2 ... cm,m | | αn |

where the matrix on the right is the matrix whose columns are the vectors in S1, and the matrix on the left is the matrix whose columns are the vectors in S2.

Since S2 is also a basis of V, the matrix on the left is invertible. Therefore, we can multiply both sides of the equation by the inverse of the matrix on the left, giving

| α1 | | b1,1 b1,2 ... b1,m | | β1 |

| α2 | | b2,1 b2,2 ... b2,m | | β2 |

| ... | = | ... ... ... ... | * | ... |

| αn | | bn,1 bn,2 ... bn,m | | βm |

where b1,j, b2,j, ..., bn,j are scalars.

Now consider the determinant of the matrix on the left-hand side of this equation. Since this matrix is obtained by multiplying the matrix whose columns are the vectors in S2 by the inverse of the matrix whose columns are the vectors in S1, its determinant is equal to the product of the determinants of these two matrices:

det([α1 α2 ... αn]) * det([β1 β2 ... βm]^-1) = det([α1 α2 ... αn] [β1 β2 ... βm]^-1)

The left-hand side is nonzero, since S1 and S2 are both bases of V and therefore their vectors are linearly independent, so the determinant of each matrix is nonzero. However, the right-hand side is zero, since the product of the two matrices on the right-hand side is the identity matrix, and the determinant of the identity matrix is 1.

This is a contradiction, so our assumption that S1 has more elements than S2 must be false. Therefore, S1 and S2 have the same number of elements, and the proof is complete.

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Answer:

2:1

Step-by-step explanation:

So, let's say we have two tigers and one bear. If we want to talk about how many tigers we have compared to how many bears we have, we can say "the ratio of tigers to bears is 2 to 1."

What this means is that for every two tigers we have, we have one bear. So if we had four tigers, we would expect to have two bears, because we have two sets of two tigers each, and each set of two tigers corresponds (to be similar to something else) to one bear.

A ratio is just a way of comparing two things, like tigers and bears, by describing how much of one thing there is compared to the other thing. So when we say the ratio of tigers to bears is 2 to 1, we're saying that we have twice as many tigers as we do bears.