Solve the system by elimination. Check your solution x+2y=7 / 13-5y=x

The coordination of point G is (3,7).

What is a slope?

A line's slope, often known as its gradient, is a numerical representation of the line's steepness and direction.

Here, we have

Given

Slope = 4

Slope of line = (y₂ - y₁)/ (x₂ - x₁) ...  (1)

F is located at (2,3), which implies x₁ = 2 and y₁ = 3

Now, substituting the value in equation 1, we get

4 = y₂ - 3 / x₂ - 2

4x₂ - y₂ = 5.... (2)

let us do the hit-and-trial method in order to find the answer.

secondly, let us take option B) 3,7

similarly, we get,

4* 3 - y₂ = 5

12 - 5 = y₂

y₂ = 7

Hence, the coordination of point G is (3,7).

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It is true that the expressions 5y + 4 − 3y = 2y + 4 are equivalent

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

5y+4−3y = 2y+4

This equation can be rewritten as follows

So, we have the following representation

5y + 4 − 3y = 2y + 4

Collect the like terms in the above equation

This gives

5y − 3y + 4 = 2y + 4

Evaluate the like terms in the above equation

This gives

2y + 4 = 2y + 4

Both sides of the equation are the same

Hence, the expressions are equivalent

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It will take 9 years for Kevin to be only 4 times as old as Daniel.

The number of years it will take for Kevin to be only 4 times as old as Daniel can be determined by using an algebraic expression

Consider x represents the number of years that will need to pass. To represent Kevin's age, the expression can be given as;

39 + x

To represent 4 times Daniel's age, the expression can be given as;

4 × (3 + x)

In other words, this equation can be represented as Kevin's age plus x years is equal to four times Daniel's age plus x years.

Since Daniel's age needs to be equal to 4 times Daniel's age, we set the expressions equal to each other:

39 + x = 4 ( 3 + x )

39 + x = 12 + 4x

39 - 12 = 4x - x

3x = 27

x = 27 ÷ 3

x = 9

Although a part of your question is incorrect, you might be referring to this question:

Kevin is 39 years old and Daniel is 3 years old how many years will it take until Kevin is only 4 times as old as daniel

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These terms represent the first four terms of the Taylor series expansion of the function f(x) around the point c. The higher order terms in the series involve higher order derivatives of f(x) at the point c.

The Taylor series is a mathematical concept that gives an approximate representation of a function as a sequence of terms determined from the function's derivatives at a particular location. It is named for mathematician Brook Taylor, who first proposed the idea in the 18th century. The Taylor series enlarges a non zero function into an infinite sum of terms, each of which is a power of the independent variable (often x) multiplied by a coefficient. The value of the function and its derivatives at the location where the expansion is centred determines each term's coefficient. The Taylor series expansion of the function f(x) at the point x=a, for example, is given by:

f(x) = f(a) + (x-a) (x-a)

(x-a)2f"(a)/2! + f'(a) + (x-a) (x-a)^3f'''(a)/3! + ...

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By using the power property, the resultant answer when we rewrite log3x9 is (B) 9log3x.

Multiply the exponents to determine the power of the power.

This is a development of the powers property.

Consider raising a number to a power and repeatedly multiplying the entire phrase by itself.

According to the power of a powerful formula, you can multiply two exponents together if one is increased to a power.

How many times to multiply two numbers together is indicated by an exponent.

For instance, 42 instructs you to multiply by 4 and 4. Alternatively, multiply four by two.

So, we have: log3x9

We have the following for logarithm properties:

loga (x ^ b) = b * loga (x)

As a result of this property, in this instance, we have:

log3 (x ^ 9) = 9log3 (x)

Therefore, by using the power property, the resultant answer when we rewrite log3x9 is (B) 9log3x.

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Correct question:
Use the power property to rewrite log3x9.

A) 3log9x

B) 9log3x

C) 3logx

D) 2 + log3x

The x and y components of net electric field at origing is [tex]E_{x}=E_{y}=(\frac{2kq}{\pi a^2})\text{ or }(\frac{q}{2\pi^2 a^2\epsilon_{o}})[/tex].

In the given question, negative charge -q is distributed uniformly around a quarter-circle of radius a that lies in the first quadrant, with the center of curvature at the origin.

We have to find the x- and y-components of the net electric field at the origin.

The figure of the given question is given below:

Linear charge density of a quarter circle is given by,

λ=(-q)/l

From the figure, the length of the quarter circle is;

l = πa/2

From the above two equations,

λ=(-q)/(πa/2)

λ=(-2q)/πa

The small amount of the charge is given by,

dq=λdl

dq=(-2q)/πa dl

Magnitude of electric field due to elemental length if given by,

dE = k|dq|/a^2

dE = k|(-2q)/πa dl|/a^2

dE = (2kq/πa^3)dl

The electric field at origin due to elemental length is given by,

[tex]\vec{dE}[/tex] = dEcosθ[tex]\hat{i}[/tex] + dEsinθ[tex]\hat{j}[/tex]

[tex]\vec{dE}[/tex] = {(2kq/πa^3)dl}cosθ[tex]\hat{i}[/tex] + {(2kq/πa^3)dl}sinθ[tex]\hat{j}[/tex]

[tex]\vec{dE}[/tex] = {(2kq/πa^3)(adθ)}cosθ[tex]\hat{i}[/tex] + {(2kq/πa^3)(adθ)}sinθ[tex]\hat{j}[/tex]

[tex]\vec{dE}[/tex] = (2kq/πa^2)cosθdθ[tex]\hat{i}[/tex] + (2kq/πa^3)sinθdθ[tex]\hat{j}[/tex]

The net electric field at origin is given by,

[tex]\vec{E}=\int\limits^{\pi/2}_0 \, \vec{dE}[/tex]

[tex]\vec{E}=\frac{2kq}{\pi a^2}(\int\limits^{\pi/2}_0 \, \cos\theta {d\theta})\vec{i}+\frac{2kq}{\pi a^2}(\int\limits^{\pi/2}_0 \, \sin\theta {d\theta})\vec{j}[/tex]

[tex]\vec{E}=\frac{2kq}{\pi a^2}(1-0)\vec{i}+\frac{2kq}{\pi a^2}(0-(-1))\vec{j}[/tex]

[tex]\vec{E}=(\frac{2kq}{\pi a^2})\vec{i}+(\frac{2kq}{\pi a^2})\vec{j}[/tex]

or

[tex]\vec{E}=(\frac{2(\frac{1}{4\pi \epsilon_{o}})q}{\pi a^2})\vec{i}+(\frac{2(\frac{1}{4\pi \epsilon_{o}})q}{\pi a^2})\vec{j}[/tex]

[tex]\vec{E}=(\frac{q}{2\pi^2 a^2\epsilon_{o}})\vec{i}+(\frac{q}{2\pi^2 a^2\epsilon_{o}})\vec{j}[/tex]

Therefore, the x and y components of net electric field at origing is;

[tex]E_{x}=E_{y}=(\frac{2kq}{\pi a^2})\text{ or }(\frac{q}{2\pi^2 a^2\epsilon_{o}})[/tex]

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The average age of all students in the college is a parameter. A parameter describes the whole populations.

Tha value of parameter or statistic can be average, percentage, and others.

A college reports that the average age of their students is 28 years old.

Is this a statistic of parameter?

The college would have access to data on all students. The data of all students in the college or the entire students is called population. So, this is a parameter.

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Answer:

24m

Step-by-step explanation:

l=8m

b=4m=w

p=2(l+b)

p=2(8+4)m

p=2(12m)

p=24m

Answer:

24 cm

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5 cm}\underline{Perimeter of a rectangle} \\\\$P=2(w+l)$\\\\where:\\ \phantom{ww}$\bullet$ $w$ is the width.\\ \phantom{ww}$\bullet$ $l$ is the length. \\\end{minipage}}[/tex]

Given values:

Substitute the given values into the formula:

[tex]\begin{aligned}P&=2(w+l)\\\implies P&=2(4+8)\\&=2(12)\\&=24\;\; \sf cm\end{aligned}[/tex]

Therefore, the perimeter of the rectangle is 24 cm.

The correct interpretation of this p-value with an alpha level of 0.05 is explain below.

P-value is the margin of error importance inside a factual speculation test, addressing the likelihood of the event of a given occasion.

We have, P-value =0.07

Level of significance = 0.05

We know that,

P-value is less than level of significance, then H® is rejected,smaller the p-value longer the evidence that you should reject the null hypothesis.

Now P-value is greater than 0.05.

With a p-value greater than 0.05, fail to reject the H®.Look at multiple comparison of each Level of significance.

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The minimum size of the group so that it guarantees to contain at least 6 students from some states in America is  251  students .

In the question ,

it is given that ,

the number of states in America = 50 states

The maximum number of cases in which there are less than 6 students from any state is when there are 5 students from each state,

which is ,

total of 5x50 = 250 students.

By Pigeon Hole Principle ,

In this, if one more student is added, the state to which that student belongs will have a total of 6 students.

So, this is minimum number required to guarantee that there are at least 6 students from some state in America is 251.

5x50 + 1 = 251

Therefore , the minimum size of the group is 251 students .

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For the given function f(t) = (2t + 1) using definition of Laplace transform the required solution is L(f(t))s = [ ( 2/s²) + ( 1/s) ].

As given in the question,

Given function is equal to :

f(t) = 2t + 1

Simplify the given function using definition of Laplace transform we have,

L(f(t))s = [tex]\int\limits^\infty_0 {f(t)e^{-st} } \, dt[/tex]

          =  [tex]\int\limits^\infty_0[2t +1] e^{-st} dt[/tex]

          = [tex]2\int\limits^\infty_0 te^{-st} + \int\limits^\infty_0e^{-st} dt[/tex]

         = 2 L(t) + L(1)

L(1) = [tex]\int\limits^\infty_0e^{-st} dt[/tex]

     = (-1/s) ( 0 -1 )

     = 1/s , ( s >  0)

2L ( t ) = [tex]2\int\limits^\infty_0 te^{-st}[/tex]

        =  [tex]2[t\int\limits^\infty_0 e^{-st} - \int\limits^\infty_0 ({(d/dt)(t) \int\limits^\infty_0e^{-st} \, dt )dt][/tex]

        =  2/ s²

Now ,

L(f(t))s = 2 L(t) + L(1)

          = 2/ s² + 1/s

Therefore, the solution of the given function using Laplace transform the required solution is L(f(t))s = [ ( 2/s²) + ( 1/s) ].

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It is not of exponential order for h(t) = et2. F is of exponential order and has order. F is piecewise continuous. No function's Laplace transform is possible. g(t) = eat, where t [0,.

Non-exponential form:

To represent a scientifically notated number as a non-exponential quantity: • Remove the exponential component of the number by moving the decimal point the same number of places as the exponent's value. The decimal is moved to the right by the same number if the exponent is positive.

Here are a few instances of functions other than exponential ones. y = 3 1 x as a result. n = 0 3 p as a result. Because y = ( 4) x. Since b = 1, y = 6 0 x.

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The vector-valued function at each given value of t  is :

a) i

b)  1/√2i+9/√2j

c)−cos(θ)i+9sin(θ)j

d)  −2sin(π/6+△t/2)sin(△t/2)i+18cos(π/6+△t/2)sin(△t/2)j

The given problem is related to trigonometric Ratios of Angle, where there are six trigonometric ratios sine, cosine, tangent, cotangent, cosecant, and secant. Since it is given that a  vector valued function:

r(t)= cos(t)i + 9 sin(t)j

r(t)=cos(t)i+9sin(t)j

so ,  r(0)=  cos(0)i+9sin(0)j = i ( sine cos0 =1 and sin0 =0)

r(π/4) =  cos(0)i+4sin(0)j = cos(π/4)i+9sin(π/4)j

           = 1/√2i+9/√2j

r(θ−π)= cos(θ−π)i+9sin(θ−π)j= −cos(θ)i+9sin(θ)j

(π/6+△t)−r(π/6)= cos(π/6+△t)i+9sin(π/6+△t)j−cos(π/6)i−9sin(π/6)j

=  cos(π/6+△t)i−cos(π/6)i+9sin(π/6+△t)j−9sin(π/6)j

= −2sin(π/6+△t/2)sin(△t/2)i+18cos(π/6+△t/2)sin(△t/2)j

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Evaluate (If possible) the vector-valued function at each given value of t. (If an answer does not exist, enter DNE.)

r(t)=cos(t)i+9sin(t)jr(t)

a)r(0)=?

b)r(π/4)=?

c)r(θ−π)=?

d)r(π/6+△t)−r(π/6)=?

There are no potential roots for this polynomial.

An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). For example, a cubic function is of the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0. It has a degree of 3 and has 3 possible roots.

We have the following polynomial -

x⁴ – 2x³ + 5x² – 7x + 9.

Plot the graph of the function. The [x] - intercepts represent the possible roots of the given polynomial. It can be seen that it does not cut the [x] - axis, hence, there are no potential roots for this polynomial.

Therefore, there are no potential roots for this polynomial.

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To determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001, we can use the alternating series estimation theorem.

A mathematical technique called the series estimation theorem is used to estimate a mathematical series' value. An endless string of integers that can be put together to produce a total is referred to as a mathematical series. According to the theory, if a series possesses a characteristic called absolute convergence, its value may be calculated by adding the first few terms rather than adding all of them together. This can be a helpful tool for estimating the value of a series without having to sum all the terms, which is sometimes difficult or even impossible. For example, the series estimation theorem could be used to approximate the value.

How to solve?

In this case, the series is alternating and decreasing, so we can use the alternating series estimation theorem to determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. We can do this by finding the first neglected term in the series that is less than 0.001, and then determining how many terms are needed to get to that point in the series.

In general, to determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001, we would need to find the first neglected term in the series that is less than 0.001 and then determine how many terms are needed to get to that point in the series. This would give us the minimum number of terms that should be used to estimate the sum of the entire series with an error of less than 0.001

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We are aware that the numbers 4, 9, 25 and 49 are among those between 1 and 100 that have precisely three elements.

Given,

What number has three components?

In fact, the squares of primes are the only positive integers with precisely three factors. For instance, 1, 3, and 9 are the factors of 9, and 1, 7, and 49 are the factors of 49.

If a given number, such as x2, is perfect square, and its square root is prime, it must have precisely three different factors.

Here,

We are aware that the numbers 4, 9, 25 and 49 are among those between 1 and 100 that have precisely three elements.

1, 2, and 4 are the four factors. 1, 3, and 9 are the factors of 9.

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Answer:

- 21

Step-by-step explanation:

consecutive odd numbers have a difference of 2 between each one

let the three consecutive odd integers be

n , n + 2 , n + 4

then

n + n + 2 + n + 4 = - 63

3n + 6 = - 63 ( subtract 6 from each side )

3n = - 69 ( divide both sides by 3 )

n = - 23

n + 2 = - 23 + 2 = - 21

n + 4 = - 23 + 4 = - 19

the three odd integers are - 23, - 21, - 19

with middle one - 21

Answer:

first integer

19

second integer

21

third integer

23

Step-by-step explanation:

The sum of the three consecutive odd integers is

63

, so your equation is:

(x)+ (x+2)+(x+4)=63

Then solve for

x:

(x)+(x+2)+(x+4)=63

x+x+2+x+4=63

3x+6=63

3x+6

−

6=63 −63x=57

3x÷3=57÷3x=19

Now that you know your first integer has a value of

19, substitute x=19 into x+2 and x+4

to find the values of the second and third integers.

Now that you know your first integer has a value of

19

Now that you know your first integer has a value of

19

, substitute

x=19 into x+2 and x+4 to find the values of the second and third integers

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The first thing a sales associate should do when a customer has a problem with a product by Thanking  the customer for bringing the problem to their  attention

A sales associate may be a sort of sales professional who works in retail whose duties are geared towards interfacing directly with prospects as they enter a store and serving as a consultative resource who helps those customers discover the specific items that best suit their needs. when a customer approach a problem, understand the problem the customer has in a detailed way and ask them for the receipts, or warranties, credit card statements, canceled checks,  contracts, invoices, or other documents in order for payment issues.

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Answer:

2,800,000 Centimeters.

Step-by-step explanation:

Multiply by 100,000. (works for any kilometers to centimeters conversion. For example, 2 kilometers is 200,000 centimeters. *2 x 100,000*)

The correct answer is the radius of the circle is 2.54648 ft.

The circumference of a circle or ellipse in geometry is known as its perimeter. That is, if the circle was opened up and straightened out to a line segment, the circumference would be said the length of the arc.

The curve length around any closed figure is more commonly referred to as the perimeter.

We know that Circumference = [tex]2\pi r[/tex] and [tex]\pi[/tex] ≈ 3.14159,

Circumference = [tex]2\pi r[/tex]

16 = [tex]2\pi r[/tex] (after substituting)

16/2[tex]\pi[/tex] = 2[tex]\pi[/tex]r/2[tex]\pi[/tex] (remove 2[tex]\pi[/tex] by dividing both sides with the mentioned term)

r = 8/[tex]\pi[/tex]

r = 8/[tex]\pi[/tex] (ft.)

[tex]\pi[/tex] ≈ 3.14159 (value of [tex]\pi[/tex])

Hence, the radius of the circle when its circumference is 16 feet is r ≈ 2.54648 ft. which is the approximate value.

Checking our work:

Circumference  = 2[tex]\pi r[/tex]

Circumference  = 2 (3.14159) (2.54648)

Circumference  = 15.999

Circumference  ≈ 16 ft.

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The equation of the line is 2x+y=10 when it that passing through the point and perpendicular to the line.

Given that,

The point is (6,-2) and the line is y=-2x+8

We have to find the equation of the line that passing through the point and perpendicular to the line.

We know that,

The equation of the line is y=mx+c

Here, y and x are the value from the point that is (6,-2)

m is the slope of the line that is -2

c  we have to find

-2=-2(6)+c

-2=-12+c

c=-2+12

c=10

The equation of the line is y=-2x+10

2x+y=10

Therefore, The equation of the line is 2x+y=10 when it that passing through the point and perpendicular to the line.

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Using Tap working problem solving,

when faucet and drain working together, the time to fill up the tub is 60 minutes.

We have the following information about question,

The time taken by faucet to fill up the tub

= 15 minutes

The time taken by drain to empty the tub

= 20 minutes

we have to calculate the time in which the tub is fully fill the tube with drain open.

Rate of fill up the tub by faucet = 1/15 i.e 1/15 th of tub in one minute .

Rate of empty the tub by drain = 1/20 i.e 1/20 th of tub in one minute.

In one minute both are working together and will fill the tub = 1/15 - 1/20 = 1/60 i.e 1/60 th of tub

Since, it takes one minute to fill 1/60th of the tub and it will take 60 minutes to fill the tub.

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The density of horses in royale stables would be 5.78 horses/sq.km  

How to find the population density?

To calculate the population density, you will divide the population by the size of the area. Thus, Population Density = Number of People/Land Area. The unit of land area should be square miles or square kilometers.

Given that there are 4564 horses in royale stables. and the stables have an area of 789 sq.km

So by using the formula for population density, we can find the population density of horses in royale stables as

Population density of horses in royale stables = number of horses in royale stables/ Area of Royale stables

                                                             = 4564 / 789

                                                             = 5.78 horses/sq.km        

Hence, the density of horses in royale stables would be 5.78 horses/sq.km  

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The sum of first 20 multiples of 7 will be 1470.

Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

Given are first 20 multiples of 7.

The sequence of first 20 multiples of 7 will be -

7, 14, 21, 28, ..... , 140

Sum = (20/2)[2 x 7 + (19) x 7]

Sum = 10[14 + 133]

Sum = 10 x 147

Sum = 1470

Therefore, the sum of first 20 multiples of 7 will be 1470.

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[Question translation in english -

find the sum of the first 20 multiples of 7 with procedure.]

The measure of both angles is 60 and 180 degrees respectively.

In a literal sense, corresponding means pairwise angles. Like the right corner angles of two triangles etc. But usually, we take them as:

For two similar figures, the pair-by-pair similar angles of those two similar figures are called corresponding angles. They are of the same measurement.

Given that angle a is corresponding to angle b the measure of angle b is 2x+60 and the measure of angle a is x.

Since both the angles are equal then;

Angle A = Angle B

x = 2x + 60

Solving for x;

2x - x = -60

x = -60

Therefore, the measure of both angles is 60 and 180 degrees.

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Using the exponential model F₀ = x, the annual rate of deforestation in El Salvador is 2.65%. .It is estimated that at the present rate of deforestation in El Salvador, in 20 years only 53%of the present forest will be remaining.

A number written as a superscript over another number is known as an exponent. In other words, it means that the base has been elevated to a particular level of power. Other names for the exponent are index and power. Mn indicates that m has been multiplied by itself n times if m is a positive number and n is its exponent.

Given,

It is estimated that at the present rate of deforestation in El Salvador, in 20 years only 53%of the present forest will be remaining.

Let the area present in forest be x,

F₀ = x,

Forest's final area:

F = 53/100 x

F = 0.53x

Time = 20 years

As we know,

Exponential model,

F = F₀ e^rt

0.53 =  e^r(20)

Equating en both sides,

en 0.53 = en e ^20r

en 0.53 = 20r

r = en 0.53/20

r = 0.0265

r - 2.65%

Using the exponential model F₀ = x, the annual rate of deforestation in El Salvador is 2.65%. .It is estimated that at the present rate of deforestation in El Salvador, in 20 years only 53%of the present forest will be remaining.

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An equation that has fractions with xs in the numerator, denominator, or both is referred to as rational. A rational equation looks like this: (4 / (x + 1)) - (3 / (x - 1)) = -2 / (x^2 - 1).

Consider the last time you used a fraction to solve an equation. 1/3 x = 8. The 3 in the denominator is a prisoner in our eyes, and we wish to free it. We multiply both sides of the equation by 3 to release the 3. Consider it as 3 notifying both parties of his departure. 3 (1/3 x) = 8 (3). (3).This procedure eliminated the fraction x = 24 and released our denominator. We add one more step to it when we need to solve rational equations. Sometimes, even though our answer to a rational equation appears to be sound, it actually contains a virus that prevents it from working. We refer to these as superfluous solutions. To answer a rational equation, follow these steps:

1. Determine the common denominator.

2. Divide all sums by the lowest common denominator.

3. Simplify.

4. Verify the solution(s) to make sure there aren't any unnecessary ones.

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Margin of error for the given standard deviation is  m = 5.497 , 5.703

What is standard deviation?

Standard Deviation is a measure that shows what quantity variation (such as unfold, dispersion, spread,) from the mean exists. the quality deviation indicates a “typical” deviation from the mean. it's a well-liked live of variability as a result of it returns to the first units of live of the info set.

Main body:

formula for confidence interval is :

m = x-bar ± z* σ/√n

given information =

n = 25

σ = 0.3

mean = 5.6 ft

C.I. = 90% = 0.9

value of 0.9 in z - table = 0.816

substituting value ,

m = 5.6 ±(0.816) * 0.3/√25

m = 5.6 ± 0.103

m = 5.497 , 5.703

Hence margin of error is m = 5.497 , 5.703

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For smallest value n=127, the decimal representation is possible.

When the first three digits after the decimal point are 0.251,

we consider the smallest value of n....

Assume the number is otherwise in the form of

[tex]\frac{m}{n} = 0.X251 \ldots,[/tex]

where X is a k-digit string and n is as small as possible.

Then

[tex]10^k \cdot \frac{m}{n} - X = \frac{10^k m - nX}{n} = 0.251 \ldots[/tex]

. Since  [tex]10^k m - nX[/tex] is an integer and [tex]\frac{10^k m - nX}{n}[/tex] is a fraction between 0 and 1, we can rewrite this as [tex]\frac{10^k m - nX}{n} = \frac{p}{q}[/tex], where [tex]q \le n[/tex]. Then the fraction [tex]$\frac pq = 0.251 \ldots$[/tex]   suffices.

Thus we have[tex]\frac{m'}{n} = 0.251\ldots[/tex], or

[tex]\frac{251}{1000} \le \frac{m'}{n} < \frac{252}{1000} \Longleftrightarrow 251n \le 1000m' < 252n \Longleftrightarrow n \le 250(4m'-n) < 2n.[/tex]

As [tex]$4m' > n$[/tex], we understand that the minimum value of [tex]$4m' - n$[/tex] is 1; hence we need [tex]$250 < 2n \Longrightarrow 125 < n$[/tex].

Since [tex]$4m' - n = 1$[/tex], we need [tex]$n + 1$[/tex] to be divisible by four, and this occurs for the first time when n = [tex]\boxed{ 127 }[/tex]

(note that if[tex]4m'-n > 1,[/tex] then [tex]$n > 250$[/tex]).

this gives [tex]$m' = 32$[/tex] and the fraction [tex]$\frac {32}{127}\approx 0.25196 \ldots$[/tex]).

Thus, the smallest value for n is 127.

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