Let a, b, and c be elements of a field F.Prove:
(1) If a + b = c + b, then a = c.
(2) If ab = cb and b doesn't equal 0, then a = c.

Yes, the pair (x, y) exists and satisfies the given equation.

Yes, there exists a pair of integers (x, y) such that the greatest common divisor (d) of a and b can be expressed as a linear combination of the two numbers, i.e. d = ax + by.

To prove this, first note that the greatest common divisor (d) of a and b divides both a and b. Therefore, by the Division Algorithm, there exist integers q and r such that a = dq + r and 0 ≤ r < d. Similarly, there exist integers p and s such that b = dp + s and 0 ≤ s < d.

Subtracting these two equations yields d = (a - dq) + (b - dp) = (r - q) + (s - p). Therefore, if we let x = r - q and y = s - p, then d = ax + by, where x and y are both integers. Thus, the pair (x, y) exists and satisfies the given equation.

Answer:

1.5

Step-by-step explanation:

See the screenshots for step by step explanation

 Use symoballab it is helpfull

The simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.

The expression (x^(2)+8x+16)/(x^(2)-4x-32) can be simplified by factoring the numerator and denominator.

First, we can factor the numerator:
(x^(2)+8x+16) = (x+4)(x+4)

Next, we can factor the denominator:
(x^(2)-4x-32) = (x-8)(x+4)

Now, we can simplify the expression by canceling out the common factor of (x+4):
(x+4)(x+4)/(x-8)(x+4) = (x+4)/(x-8)

Therefore, the simplest form of the expression is (x+4)/(x-8).

However, there are restrictions on the variable x. The denominator of the expression cannot equal zero, so we must find the values of x that make the denominator zero and exclude them from the domain of the expression.

To find the restrictions, we set the denominator equal to zero and solve for x:
(x-8)(x+4) = 0

This gives us two solutions:
x = 8
x = -4

Therefore, the restrictions on the variable are x≠8 and x≠-4.

In conclusion, the simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.

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For part (a), the answer is f(x+h) = x² + 2xh + h² - 4x - 4h. for par (b) h = -x ± sqrt(x² + 6), the value of h depends on the value of x

(a) To find f(x+h), we substitute x+h for x in the expression for f(x):

f(x+h) = (x+h)² - 4(x+h)

Expanding the square and simplifying, we get:

f(x+h) = x² + 2xh + h² - 4x - 4h

(b) To find h, we start with the expression for f(x+h) that we found in part (a):

f(x+h) = x² + 2xh + h² - 4x - 4h

We want to simplify this expression so that we can identify h. To do this, we start by subtracting f(x) from both sides:

f(x+h) - f(x) = (x² + 2xh + h² - 4x - 4h) - (x² - 4x)

Simplifying, we get:

f(x+h) - f(x) = 2xh + h² - 4h

Now we can identify h by setting this expression equal to some value and solving for h. For example, if we set f(x+h) - f(x) equal to 5, we get:

2xh + h² - 4h = 5

Simplifying and rearranging, we get a quadratic equation in h:

h² + 2xh - 4h - 5 = 0

We can solve this using the quadratic formula:

h = (-2x ± √(4x² + 24))/2

Simplifying, we get:

h = -x ± √(x² + 6)

So the value of h depends on the value of x

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he correct answer is d. 4/ x^2 -2x+1 + 1.

To find f(g(x)), we need to substitute g(x) into f(x).

f(x) = x^2

g(x) = 2/x-1

So, f(g(x)) = (2/x-1)^2

= 4/x^2 - 4/x + 1

Therefore, the correct answer is d. 4/ x^2 -2x+1 + 1.

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Answer: Your welcome!

Step-by-step explanation:

The students can decorate the circles for placemats in a variety of ways. They can use paint, markers, fabric, or any other creative material of their choice. They can also add images, shapes, and words to the circles. They could even attach ribbons or other decorations to the circles to create a unique design. The possibilities are endless!

The correct answer is option A. 495, -305.

To find a positive angle that is coterminal to the given angle, we can add 360 degrees to the given angle. This is because coterminal angles are angles that have the same initial side and terminal side, but differ in the number of rotations. Adding 360 degrees to the given angle gives us the same initial side and terminal side, but with one extra rotation.

95 + 360 = 455

To find a negative angle that is coterminal to the given angle, we can subtract 360 degrees from the given angle. This gives us the same initial side and terminal side, but with one less rotation.

95 - 360 = -265

Therefore, the positive angle that is coterminal to the given angle is 455, and the negative angle that is coterminal to the given angle is -265.

Answer: A. 495, -305.

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The rate of change is 961.37 gallons/hour. This means that the water level decreases by 961.37 gallons every hour.

In mathematics, comparing two related quantities stated in different units is known as a rate. A rate is the ratio of two distinct values that are gauged in various ways. It is not the same as a unit rate to compare a certain number of units of the first quantity to one unit of the second. In general, the rate can be expressed as the ratio of two quantities with distinct units. With the help of rate, we can create a connection between two otherwise unconnected components. By doing this, we can gain a deeper understanding of a scenario.

Given,

The total amount of water the pool can hold =  9613.7 gallons

The time taken to drain the pool completely = 10 hours

We are asked to find the change in water level per hour of the pool.

This can be called the rate of change.

Rate of change = Total amount of water/ Time taken to drain

                       = 9613.7 / 10 = 961.37 gallons / hour

Therefore the rate of change is 961.37 gallons/hour. This means that the water level decreases by 961.37 gallons every hour.

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The variance of the error term is equal to the expected value of the squared error term, which is equal to the variance of Xi.

The asymptotic distribution of the coefficient β from bivariate OLS is given by √N(β – β) + d(0,σ2), where σ2 is the variance of the error term and is given by Var((Xi - E[X])) / Var(xi)2. If the error term is homoskedastic, then the variance of the error term does not depend on the value of Xi and is constant for all values of Xi. This implies that the variance of Y given X and r does not depend on r, as shown below:

Var(Y|X, r) = Var(βX + ε|X, r) = Var(ε|X, r) = σ2

Since the variance of the error term is constant and does not depend on the value of X or r, the variance of Y given X and r is also constant and does not depend on r.

If Yi is earnings and Xi is an indicator for whether someone has gone to college, homoskedasticity would imply that the variance of earnings for college and non-college workers is the same. This is unlikely to hold in practice, as there are likely to be other factors that affect earnings, such as occupation, experience, and location, that may differ between college and non-college workers and lead to different variances in earnings.

If the error term is homoskedastic and E[εi|Xi] = 0, then the variance of the error term is equal to the variance of Xi, as shown below:

Var(εi) = E[εi2] - (E[εi])2 = E[εi2] = E[(Xi - E[Xi])2] = Var(Xi)

This is because the error term is uncorrelated with Xi and has a mean of zero

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The fraction of total distance that was travelled on Sunday is 3/8.

Let total distance to Fernando's grandmother's house be = D;

On Saturday, they traveled 3/8 of D,

So, the remaining distance is 5/8 of D.

On Sunday, they traveled 3/5 of the remaining distance, which is:

⇒ (3/5) × (5/8) × D = 3/8 × D,

So, on Sunday, they traveled 3/8 of D.

To find the fraction of the total distance traveled on Sunday, we divide the distance traveled on Sunday by the total distance, which is;

⇒ (3D/8)/D = 3/8

Therefore, Fernando's family traveled 3/8 of the total distance on Sunday.

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The given question is incomplete, the complete question is

Fernando's family traveled 3/8 of the distance to his grandmother’s house on Saturday. They traveled 3/5 of the remaining distance on Sunday.

What fraction of the total distance to his grandmother’s house was traveled on Sunday?

the factored form of the expression 9m^(2)-49 is (3m+7)(3m-7).

To factor the expression 9m^(2)-49, we need to use the difference of squares formula.

The difference of squares formula states that a^(2)-b^(2)=(a+b)(a-b).

In this case, we can see that 9m^(2) is a perfect square and 49 is also a perfect square. So, we can write the expression as (3m)^(2)-(7)^(2).

Using the difference of squares formula, we get:

(3m)^(2)-(7)^(2)=(3m+7)(3m-7)

So, the factored form of the expression 9m^(2)-49 is (3m+7)(3m-7).

Therefore, the answer is (3m+7)(3m-7).

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The answers are:

35. f(x)⋅g(x) = 9x2 - 0.25

36. (f(x))2 = 9x2 + 3x + 0.25

37. (g(x))2 = 9x2 - 3x + 0.25

To MULTIPLYING FUNCTIONS, we simply multiply the corresponding terms of each function together. Let's use the given functions f(x)=3x+0.5 and g(x)=3x−0.5 and perform the indicated operations.

35. f(x)⋅g(x) = (3x+0.5)(3x−0.5) = 9x2 - 0.25

36. (f(x))2 = (3x+0.5)2 = 9x2 + 3x + 0.25

37. (g(x))2 = (3x-0.5)2 = 9x2 - 3x + 0.25

So, the answers are:

35. f(x)⋅g(x) = 9x2 - 0.25

36. (f(x))2 = 9x2 + 3x + 0.25

37. (g(x))2 = 9x2 - 3x + 0.25

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Answer:

Step-by-step explanation:

Let x be the first term.  Let y be the common difference between each number in the sequence. x and the next three terms would be:

x, x+y,  x+2y,  and x+3y

The sum of the 4 terms is 4x + 6y and is equal to 38

4x + 6y = 38

4x  = 38 - 6y

x  = (19/2) - (3/2)y   [x is isolated here, to the left, for use in a lovely substitution coming up]

or x = 9.5 - 1.5y  [simplified]

===

The sum of the first 7 terms would be the first 4 [from above:   4x + 6y] plus the next 3 terms;

4x + 6y

 x + 4y

 x + 5y

 x + 6y

7x + 21y

7x + 21y is equal to 98

7x + 21y = 98

====

We have two equations and two unknowns, so we should be able to find an answer by substitution:

---

From above:

x  = (19/2) - (3/2)y  

7x + 21y = 98

Now use the first definition of x in the second equation:

7x + 21y = 98

7( (19/2) - (3/2)y) + 21a = 98

66.5 - 10.5y + 21y = 98

10.5y = 31.5

y = 3

Now use this value of y in either equation to find x:

7x + 21*(3) = 98

7x + 63 = 98

7x = 35

x =  5

====

Check:

Do the first 4 terms sum to 38?

5 + 8 + 11 + 14 = 38  YES

Do the first 7 terms sum to 98?

38 + 17 + 20 + 23 =   YES

Answer:

35.13

Rounded: 35.1

Step-by-step explanation:

The figure will look like the attached image.

(My guess, based on hint)

We will first find the area of the circle.

Area of circle is radius squared times Pi

3x3x3.14=28.26

It is half circle so 28.26/2=14.13

The question didn't state the height, but the base is 6 and we'll assume it is 7 because the question has 7 in the third to last row. 7x6/2=21

14.13+21=35.13

a)2

b)4

c)6

d)x2

e)-2a2b

For questions 7, 8, and 10:

The cube root of 7 is 2, the cube root of 40 is 4, and the cube root of 162 is 6. The cube root of an expression with an exponent can be found by dividing the exponent by 3; for example, the cube root of x8 is x2.

For question 11:

The cube root of -16a5b is -2a2b.

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The given polynomial g(x) is g(x)=x^(3)+(p-4)x^(2)+(p-9)x-4.

To show that the polynomial g(x) is divisible by (x+1) for all values of p, we must prove that g(x) has a factor of (x+1). This can be done by expanding the given polynomial and rewriting it in a form that contains the factor (x+1).

First, we expand the given polynomial g(x):

g(x)=(x^3 + (p-4)x^2 + (p-9)x - 4)

= x^3 + px^2 - 4x^2 + px - 9x - 4

= x^3 + (p-4)x^2 + (px - 9x) - 4

= x^3 + (p-4)x^2 + (p - 9)x - 4

Now we can factor out the term (x+1) from the expression:

= (x^3 + (p-4)x^2 + (p-9)x - 4)

= (x+1)(x^2 + (p-5)x + (p-9)) - 4

Finally, we can rewrite the expression as:

g(x)=(x+1)(x^2 + (p-5)x + (p-9)) - 4

This shows that the polynomial g(x) is divisible by (x+1) for all values of p, as there is a factor of (x+1) in the expression.

Therefore, we have successfully proved that the polynomial g(x) is divisible by (x+1) for all values of p.

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The measures of angles 4 and 5 are:

∠5 = 104°

∠4 =76°

On the image we can see that angles 1 and 2 are next to eachother, that means that the sum of their measures must be a plane angle, that is an angle of 180°.

Then we can write the sum:

∠1 + ∠2 = 180°

(3x + 5) + (2x + 10) = 180

5x + 15 = 180

5x = 180 - 15

x = 165/5 = 33

the measure of angle 5 is the same one of angle 1 then:

∠5 = (3*33 + 5)° = 104°

And angle 4 is supplementary of angle 1, then:

∠4 + 104° = 180°

∠4 = 180 - 104= 76°

These are the measures

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The closest distance that a cinema can put seats is 10.667 m.

One of the six mathematical functions used to express the side ratios of right triangles, the trigonometric function includes the sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). The figure shows these six trigonometric functions in respect to a right triangle. Surveying, engineering, and navigation issues where one of a right triangle's acute angles and the length of a side are known but the lengths of the other sides need to be determined can be solved with ease using trigonometry.

As per the given data:

the angle of elevation from a customer's eyes

to the top of the screen ≤ 31°.

Top of the cinema screen = 7.6 m above floor.

Customers' eyes =  1.2 m above floor.

For the closest distance that a cinema can put the seats (B):

the angle of elevation from a customer's eyes

to the top of the screen (maximum) = 31°.

Distance between customer eyes and top of the screen (P) = 7.6 - 1.2 = 6.4 m

∴ With θ = 31° , tanθ = (P/B)

tan31° = (6.4/d)

d = (6.4/tan31°)

d = 10.667 m

Hence, the closest distance that a cinema can put seats is 10.667 m.

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a) The inverse sine of 1 is π/2 radians, which is equivalent to 90 degrees.

b) The inverse sine of √2 / 2 is π/4 radians, which is equivalent to 45 degrees.
c) The inverse sine of - √2 / 2 is -π/4 radians, which is equivalent to -45 degrees.

Evaluate the following expressions. Your answer will be an angle in radians and in the interval [-π/2, π/2]
(a) sin^-1 (1) = π/2
(b) sin^-1 (√2 / 2)= π/4
(c) sin^-1 (- √2 / 2)= -π/4


(a) sin^-1 (1) = π/2
The inverse sine of 1 is π/2 radians, which is equivalent to 90 degrees.

(b) sin^-1 (√2 / 2)= π/4
The inverse sine of √2 / 2 is π/4 radians, which is equivalent to 45 degrees.

(c) sin^-1 (- √2 / 2)= -π/4
The inverse sine of - √2 / 2 is -π/4 radians, which is equivalent to -45 degrees.

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The area of the sector outlined by the given arc is 32π square units.

What is the area of the sector?

The area of a sector of a circle with radius r and central angle θ (in radians) is given by the formula:

sector area = (1/2) x r² x θ

The length of an arc of a circle with radius r and central angle θ (in radians) is given by the formula:

arc length = r x θ

In this case, we know that the radius is 8 units and the arc length is 4π units. So we can set up an equation:

4π = 8θ

Solving for θ:

θ = (4π)/8 = π/2

So the central angle is π/2 radians.

The area of a sector of a circle with radius r and central angle θ (in radians) is given by the formula:

sector area = (1/2) x r² x θ

Plugging in the values we know:

sector area = (1/2) x 8² x π/2

sector area = 32π

Therefore, the area of the sector outlined by the given arc is 32π square units.

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Equation Slope-Intercept Form Slope y-Intercept

y = 2x + 5 y = 2x + 5 2 5

4x - 3y = 12 y = (4/3)x - 4 4/3 -4

x = 7 Does not exist Does not exist Does not exist

3y = 9 y = 3 0 3

For the first equation, y = 2x + 5, it is already in slope-intercept form. The slope is the coefficient of x, which is 2, and the y-intercept is the constant term, which is 5.

For the second equation, 4x - 3y = 12, we can rearrange it to get it in slope-intercept form by isolating y on one side of the equation. We can do this by subtracting 4x from both sides and then dividing by -3: 4x - 3y = 12 -3y = -4x + 12 y = (4/3)x - 4

The slope is the coefficient of x, which is 4/3, and the y-intercept is the constant term, which is -4. For the third equation, x = 7, there is no y term, so it cannot be written in slope-intercept form and the slope and y-intercept do not exist.

For the fourth equation, 3y = 9, we can rearrange it to get it in slope-intercept form by dividing both sides by 3: 3y = 9 y = 3 The slope is 0, since there is no x term, and the y-intercept is the constant term, which is 3.

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Simplifying the equation of v 4. 4 ( v - 16. 8 ) - -2. 3 = 3. 62, we find that v = 21.45.

To solve for v, we will use the following steps:

Simplify the left-hand side of the equation by distributing the 4.4:

4.4v - 73.92 + 2.3 = 3.62

Simplify further by combining like terms:

4.4v - 71.62 = 0

Add 71.62 to both sides of the equation:

4.4v = 71.62

Solve for v by dividing both sides by 4.4:

v = 71.62 / 4.4

v = 16.27727...

Round the answer to two decimal places:

v = 21.45

Therefore, the solution to the equation 4.4(v - 16.8) - (-2.3) = 3.62 is v = 21.45.

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You can make 16 headbands using the blue fabric and 5 headbands using the yellow fabric, for a total of 21 headbands.

To calculate this, we need to convert 2/3 yard to feet since the fabric is given in feet. 2/3 yard is equal to 2 feet.

Next, we need to determine how much fabric is needed to make one headband, which is 2 feet.

To find out how many headbands we can make using the blue fabric, we divide the total length of blue fabric (12 feet) by the length needed for one headband (2 feet). 12 feet / 2 feet = 6 headbands per yard of blue fabric. Therefore, with 12 feet of blue fabric, we can make 6 x 12 = 72 headbands.

Similarly, to find out how many headbands we can make using the yellow fabric, we divide the total length of yellow fabric (4 feet) by the length needed for one headband (2 feet). 4 feet / 2 feet = 2 headbands per yard of yellow fabric. Therefore, with 4 feet of yellow fabric, we can make 2 x 4 = 8 headbands.

Therefore, we can make a total of 72 + 8 = 80 headbands.

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Complete Question:

You need 2/3 yard of fabric to create a headband: You have 12 feet of blue fabric and 4 feet of yellow fabric; How many headbands can you make with all of the fabric?

Answer: The factor that does not influence interest rates in regards to loans is "Number of Children". The other options, collateral, credit history, and inflation, can all have an impact on interest rates. Collateral refers to the assets that a borrower pledges as security for the loan, and the value of these assets can affect the interest rate. Credit history refers to a borrower's past performance in repaying debts, which can affect their perceived risk and therefore the interest rate they are offered. Inflation refers to the general increase in prices over time, and can affect interest rates because lenders will want to charge a rate that compensates them for the loss of purchasing power due to inflation. However, the number of children a borrower has is not a factor that lenders consider when setting interest rates.

Step-by-step explanation:

The integral of f(x) from a=-1 to b=4 is \(\frac{21}{64}\). The lower and upper limits of integration are the same, the integral is 0.

a) To evaluate the integral of \(f(x)=\frac{1}{x^4}\) from a=-1 to b=4, we need to first find the antiderivative of f(x). The antiderivative of \(\frac{1}{x^4}\) is \(-\frac{1}{3x^3}\). Now we can use the Fundamental Theorem of Calculus to evaluate the integral:

Integral [a=-1,b=4] f(x) dx = \(-\frac{1}{3(4)^3} - (-\frac{1}{3(-1)^3})\)

= \(-\frac{1}{192} + \frac{1}{3}\)

= \(\frac{63}{192}\)

= \(\frac{21}{64}\)

So the integral of f(x) from a=-1 to b=4 is \(\frac{21}{64}\).

b) The integral of f(x) from a=4 to b=4 is 0. This is because the integral of a function over an interval of length 0 is always 0. In other words, if the lower and upper limits of integration are the same, the integral is 0.

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Answer: y = 4 + 3x

Step-by-step explanation:

Input (x) = any number

Output (y) = 4 more than 3 times x

more: +

times: ×

(a) Possible number(s) of positive real zeros: 0, 1, or 2
(b) Possible number(s) of negative real zeros: 0, 1, or 2


To determine the possible number of positive and negative real zeros of a polynomial function, we can use the Descartes' Rule of Signs.

This rule states that the number of positive real zeros of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial, or less than that by an even number. Similarly, the number of negative real zeros of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial when the variable x is replaced with -x, or less than that by an even number.

For the given polynomial function Q(x) = -x⁴ - 6x³ - 8x² - 5x + 1, the number of sign changes in the coefficients is 1 (from -5x to +1). Therefore, the possible number of positive real zeros is 1 or 0 (1-2).

To find the possible number of negative real zeros, we replace x with -x and simplify the polynomial:

Q(-x) = -(-x)⁴ - 6(-x)³ - 8(-x)² - 5(-x) + 1
= -x⁴ + 6x³ - 8x² + 5x + 1

The number of sign changes in the coefficients of this polynomial is 2 (from -x⁴ to +6x³ and from -8x² to +5x). Therefore, the possible number of negative real zeros is 2 or 0 (2-2).

So, the possible number of positive real zeros is 0, 1, or 2, and the possible number of negative real zeros is 0, 1, or 2.

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Answer:

4:3

Step-by-step explanation:

16:12=8:6=4:3, that is simplest ratio

Answer:

C but it's a guess........

Answer:

he walks 300 meters

Step-by-step explanation:

nbsjdjsj d f f f f f f f. f f f f f f f f f f