Solve the following system of equations. fill in the x-value and the y-value in the ordered pair below. y = 3x-5 2x-5y=25​

Answer:

up down

Step-by-step explanation:

The standard deviation of the difference Y - X depends on the standard deviations of both X and Y, and is given by the formula below.

(b) (i) The mean of the difference Y - X is μY - μX, which is 0 since both X and Y have the same mean, μ.

(b) (ii) The standard deviation of the difference Y - X is √(σY^2 + σX^2), where σY and σX are the standard deviations of Y and X, respectively. Assuming that the values of X are independent of the value of Y, we can use the formula for the variance of the difference of two random variables to get:

Var(Y - X) = Var(Y) + Var(X) - 2Cov(Y, X)

where Cov(Y, X) is the covariance between Y and X. Since Y and X have the same mean, we have:

Cov(Y, X) = E[(Y - μ)(X - μ)] = E[(Y - μ)X] - μE[X] = E[(Y - μ)X] - μ^2

where E[] denotes the expected value. Since X and Y are independent, we have:

E[(Y - μ)X] = E[Y - μ]E[X] = 0

Therefore, we get:

Var(Y - X) = Var(Y) + Var(X) = σY^2 + σX^2

Taking the square root of both sides gives us the standard deviation of Y - X:

SD(Y - X) = √(σY^2 + σX^2)

Therefore, the standard deviation of the difference Y - X depends on the standard deviations of both X and Y, and is given by the formula above.

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Answer:

a. To find the number of diagonals in an n-sided polygon, we need to count the number of pairs of vertices that can be connected by a diagonal, while excluding the sides of the polygon. Any pair of vertices that are adjacent or separated by only one vertex are already connected by a side of the polygon, so we need to consider pairs of vertices that are separated by at least two vertices.

We can choose two vertices out of n vertices in the polygon in nC2 ways. However, any pair of vertices that are separated by only one vertex defines a side of the polygon, so we need to subtract n from this count to exclude the n sides of the polygon. Therefore, the number of diagonals in an n-sided polygon is:

nC2 - n = (n * (n-1))/2 - n = (n^2 - 3n)/2

b. The above expression gives the number of diagonals for any polygon, not just a convex one. However, for a convex polygon, any pair of vertices separated by only one vertex defines a side of the polygon, so we need to exclude these diagonals as well.

For an n-sided convex polygon, there are n sides and n vertices, so we have to exclude n diagonals (one diagonal for each vertex). Therefore, the number of diagonals in an n-sided convex polygon is:

(n^2 - 3n)/2 - n = (n^2 - 5n)/2

Step-by-step explanation:

Using trigonometric functions, we can find the value of the angles as:

Sinθ = 3/5

Cosθ = 4/5

Tanθ = 3/4

Six basic trigonometric operations are used in trigonometry. These operations are described by trigonometric ratios. The six basic trigonometric functions are the sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function. The foundation for trigonometric identities and functions is the ratio of the sides of a right-angled triangle. The sine, cosine, tangent, secant, and cotangent values are computed for the perpendicular side, hypotenuse, and base of a right triangle using trigonometric formulas.

Here as per the question,

Opposite side of the angle = 3 units.

Adjacent side of the angle = 4 units.

Hypotenuse of the triangle = 5 units.

Sinθ = opposite side/ hypotenuse

= 3/5

Cosθ = adjacent side/ hypotenuse

= 4/5

Tanθ = opposite side/adjacent side

= 3/4

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The amplitude is 2 units.

The period is 2π units.

f(x) = 2 sin(x - π/2)

An oscillation that repeats itself repeatedly in many natural events is described by the sine function, which is a mathematical function. It is described as the proportion of a right triangle's hypotenuse to the side that faces an angle. The typical notation for the sine function is sin(x), where x represents the angle.

Amplitude:

The distance between the graph's midline and its maximum or minimum point is known as the amplitude. The midline in the following graph is located at y=0, midway between the greatest value of 2, and the minimum value of -2. The amplitude is therefore 2 units.

Period:

The distance among two consecutive graph points with the same value and direction is known as the period. The above graph repeats itself every 2π units of x and has one complete cycle between x=0 and x=2π. The duration is therefore 2π units.

Midline:

The midline, also referred to as the average of the highest and minimum values, traverses the graph's centre horizontally. The x-axis or the line y=0 serves as the midline in the presented graph.

It is possible to write the sine function equation for the provided graph with a horizontal shift between 0 and 2 as:

f(x) = 2 sin(x - π/2)

The graph fluctuates up and down near the midline, hence the sine function is utilised. The highest value occurs at x=π/2, which is π/2 units to the right of x=0, and the graph continues every 2π units of x, therefore the horizontal shift is π/2. The sine function multiplies the amplitude of two units.

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Step-by-step explanation:

If Debbie saves $900 each summer for four summers, she will have saved a total of:

$900 x 4 = $3600

Her parents have agreed to pay half the tuition, which is $8,500 / 2 = $4250.

Therefore, Debbie will need to come up with the remaining amount:

$8,500 - $4,250 = $4,250

Since Debbie will have saved $3,600 from her summer job, she will need an additional:

$4,250 - $3,600 = $650

Debbie will need $650 more to afford her first year's tuition.

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The given function is not one-to-one.

To determine whether the function y = 2(x+1)²-4 is one-to-one, we need to check whether each unique value of x produces a unique value of y.

Let's begin by assuming that two different values of x, say x1 and x2, produce the same value of y, i.e., y(x1) = y(x2).

Then we have:

2(x1+1)²-4 = 2(x2+1)²-4

Simplifying this equation, we get:

(x1+1)² = (x2+1)²

Taking the square root of both sides, we get:

x1+1 = ±(x2+1)

We can now solve for x1 in terms of x2 as follows:

x1 = -(x2+1)±1

Since there are two possible values for x1 corresponding to a given value of x2, the function is not one-to-one.

Therefore, the given function is not one-to-one.

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Answer:

x > 35

Step-by-step explanation:

Let's denote the width of the picture frame as "x" inches.

According to the problem, the length of the picture frame is 7 inches more than the width, so it can be expressed as "x + 7" inches.

The perimeter of the picture frame can be found by adding up the lengths of all four sides, which is given by:

Perimeter = 2*(length + width)

Substituting the values of length and width in terms of x, we get:

Perimeter = 2*(x + 7 + x)

Perimeter = 4x + 14

Now we can set up an inequality to find the values of x for which the perimeter is greater than 154 inches:

4x + 14 > 154

Subtracting 14 from both sides, we get:

4x > 140

Dividing both sides by 4, we get:

x > 35

Therefore, the values of x for which the perimeter of the picture frame is greater than 154 inches are x > 35.

Answer:

Hurit Imala's gross annual pay is $17,680

Step-by-step explanation:

Hurit Imala works 40 hours a week, so her weekly pay would be:

40 hours/week x $8.50/hour = $340/week

To find her annual pay, we need to multiply her weekly pay by the number of weeks in a year:

$340/week x 52 weeks/year = $17,680/year

Therefore, Hurit Imala's gross annual pay is $17,680.

Answer:

Step-by-step explanation:

Solution:

A four-digit number 4ab5 is divisible by 55. Then the value of b - a is 1.

 π/30 R is the radian measure of the central angle.

What is a circle, exactly?

The collection of all points in the plane that make up a circle are all equally spaced from a certain point known as the "centre," making the form a closed two-dimensional shape.

                       The reflection symmetry line is created by all lines that traverse the circle. Additionally, it is symmetrical in rotation around the centre at all angles.

Total circle circumference would be  π d  =   28 π

this arc is a fraction of the total

7π/15 / 28π  x  2π   =   π/30 R

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The five number summary, based on the given box plot would be:

Minimum value is the left-most point or the left end of the left whisker of the box plot. In the box plot, the minimum value is 1. Q1 is the left edge of the box. It represents the value below which 25% of the data falls.

Median is the line inside the box that divides it into two parts. It represents the value below which 50% of the data falls, or the 50th percentile. The median is 7. Q3 is the right edge of the box. It represents the value below which 75% of the data falls. In your given summary, Q3 is 9.

Maximum value is the right-most point or the right end of the right whisker of the box plot.

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Answer:

Step-by-step explanation:

9x-3=4x+7

             +3

9x=4x+10

-4x=-4x

5x=10

divide the 5 by 10 which equals 2.

x=2

The test statistics do not lie in the rejection so the null hypothesis does not reject.

Here we have the following information:

[tex]\bar{x}_{1}=25.3, n_{1}=25, s_{1}=9, \bar{x}_{2}=20.5, n_{2}=23, s_{2}=8[/tex]

Here degree of feedom will be

[tex]df=n_{1}+n_{2}-2=25+23-2=46[/tex]

The critical values of t are: +/- 2.687

Rejection region:

If t < -2.687 and t > 2.687, reject H0

Pooled variance is :

[tex]s^{2}_{p}=\frac{(n_{1}-1)s_{1}^{2}+\left ( n_{2}-1 \right )s_{2}^{2}}{n_{1}+n_{2}-2}=72.8696[/tex]

The standard error is

[tex]s_{M_{1}-M_{2}}=s_{p}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}=2.4664[/tex]

and t-statistics will be

[tex]t=\frac{\bar{x}_{1}-\bar{x}_{2}}{s_{p}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}=1.946[/tex]

The test statistics do not lie in the rejection so the null hypothesis does not reject.

You cannot conclude...

not

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If Julian and two of his friends are going to share 1/4 of a pizza, then they will be sharing the pizza equally among three people. To find out how much each person will get, you need to divide the 1/4 of the pizza by 3, which gives:

1/4 ÷ 3 = 1/4 × 1/3 = 1/12

So each person will get 1/12 of the pizza.

The simplified expression is 84/125.

An expression is a combination of numbers, variables, and mathematical symbols that represent a mathematical relationship or formula. Expressions can be simple or complex and can include operations such as addition, subtraction, multiplication, and division, as well as exponents, radicals, and functions.

In the given question,

To simplify the expression 5x1 + 6x1/10 + 7x1/100 + 2x1/1,000, we can start by converting all the fractions to have a common denominator of 1,000. This gives:

5x1 + 6x100/1,000 + 7x10/1,000 + 2x1/1,000

Next, we can add the terms that have the same denominator:

(6x100 + 7x10 + 2x1)/1,000

Simplifying the numerator, we get:

(600 + 70 + 2)/1,000

Adding the terms in the numerator, we get:

672/1,000

Simplifying the fraction, we can divide both the numerator and denominator by the greatest common factor, which is 8:

84/125

Therefore, the simplified expression is 84/125.

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Answer:

The pages that will be favored for the search terms Chevrolet AND Corvette are Pages that include Chevrolet Corvette.

Step-by-step explanation:

I hope it helps:)

Where the daily growth factor is 1.15 and the starting height in inches is 7.

a. Given by, the function that calculates the height of the beanstalk in relation to the time since Gregory planted it.

[tex]f(t) = 7(1.15)^t[/tex]

where the daily growth factor is 1.15 and the starting height in inches is 7.

B). The following ratios can be calculated using the formula from section (a):

f(1)/f(0) = (71.15¹)/(71.15⁰) = 1.15

f(2)/f(1) = (71.15²)/(71.15¹ = 1.15

f(1)/f(1) = (71.15¹)/(71.15¹) = 1

[tex]f(9.56)/f(8.56) = (71.15^{9.56})/(71.15^{8.56})[/tex] ≈ 1.15

[tex]f(8.56)/f(8.55) = (71.15^{8.56})/(71.15^{8.55})[/tex] ≈ 1.15

c. For the any value of z, we have:

[tex]f(z+1)/f(z) = (71.15^{(z+1)})/(71.15^z)[/tex] = 1.15

and

[tex]f(z)/f(z-1) = (71.15^z)/(71.15^{(z-1)})[/tex] = 1.15

For question 6:

a. As a increases throughout the function's domain, f(z) [Select an answer]

Since the base of the exponential function is less than 1 (0.69 < 1), as x increases, f(x) will approach 0.

b. The 1-unit growth factor of f is

The 1-unit growth factor for f is 0.69, since f(x+1)/f(x) = 0.69.

c. The -unit growth factor of f is

The 1/2-unit growth factor for f is (0.69)¹/₂, since f(x+1/2)/f(x) = (0.69)¹/₂.

d. The 6-unit growth factor of f is

The 6-unit growth factor for f is (0.69)⁶, since f(x+6)/f(x) = (0.69)⁶.

e. The 6-unit percent change of  f is

The 6-unit percent change for f is ((0.69)^6 - 1) * 100%, which is approximately -38.77%.

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Answer:

  θ ∈ {π/4, π/2, 3π/4, 5π/4, 3π/2, 7π/4}

Step-by-step explanation:

You want the solutions to sin(2θ)sin(θ)=cos(θ) on [0, 2π).

We can use the identity for sin(2θ) to write the equation in terms of sine and cosine.

  (2sin(θ)cos(θ))·sin(θ) = cos(θ) . . . . replace sin(2θ)

  2sin(θ)²cos(θ) - cos(θ) = 0 . . . . . . . subtract cos(θ)

  cos(θ)(2 sin(θ)² -1) = 0

The zero product rule tells you a product will be zero only when one or more factors is zero. The zeros of the factors are ...

  cos(θ) = 0   ⇒   θ = π/2, 3π/2

  2sin(θ)² -1 = 0   ⇒   sin(θ) = ±√(1/2)   ⇒   θ = π/4, 3π/4, 5π/4, 7π/4

The solutions are θ ∈ {π/4, π/2, 3π/4, 5π/4, 3π/2, 7π/4}.

__

Additional comment

The solutions are the x-intercepts of the graph of sin(2θ)sin(θ) -cos(θ), the values of x where this expression is zero. A graphing calculator finds them nicely.

B is not the multiplicative inverse of A.

The multiplicative inverse matrix, also known as the inverse matrix or simply the inverse, is a matrix that when multiplied by a given matrix, results in the identity matrix. The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.

Here we have

[tex]A = \left[\begin{array}{cc}3&-2&\\0&-3&\end{array}\right][/tex]    [tex]B = \left[\begin{array}{cc}-1&2&\\0&-4&\end{array}\right][/tex]  

Product of A and B

[tex]AB = \left[\begin{array}{cc}3&-2&\\0&-3&\end{array}\right] \times \left[\begin{array}{cc}-1&2&\\0&-4&\end{array}\right][/tex]

[tex]AB = \left[\begin{array}{cc}3(-1) + (-2)(0) &3(2) + (-2)(-4)&\\0 + 0 &0 +(-3)(-4)&\end{array}\right][/tex]  

[tex]AB = \left[\begin{array}{cc}-3 &14&\\0 &12&\end{array}\right][/tex]  

Product of B and A

[tex]BA = \left[\begin{array}{cc}-1&2&\\0&-4&\end{array}\right] \times \left[\begin{array}{cc}3&-2&\\0&-3&\end{array}\right][/tex]

[tex]BA = \left[\begin{array}{cc}1(3) + 2(0)&-1(-2)+2(-3)\\0(3)+0 &0(-2)+(-4)(-3) \end{array}\right][/tex]

[tex]BA = \left[\begin{array}{cc}-3&-4\\0&12\end{array}\right][/tex]

Here AB ≠ BA ≠ I

Therefore,

B is not the multiplicative inverse of A.

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Complete Question given in the picture

The answer is: f has a local maximum at x = 2 and a local minimum at x = 4.

A local minimum is a point on the graph of a function where the function takes on its smallest value in some small interval around the point. More formally, a point (x,y) is a local minimum if there exists some open interval (a,b) containing x such that for all values of x in (a,b), y is greater than or equal to f(x). Visually, a local minimum is a "valley" on the graph of the function where the function is lower than all nearby points, but may still be higher than points farther away.

In the given question,

(a)

The curve labeled y = f'(x) represents the derivative of a function f(x). Since f'(x) is positive when f(x) is increasing and negative when f(x) is decreasing, we can use the sign of f'(x) to determine the intervals of increase and decrease of f(x).

From the given information, we can sketch the graph of f'(x) as follows:

Based on this graph, we can see that f(x) is increasing on the intervals: [0, 1) and (3, 5).

Similarly, f(x) is decreasing on the intervals: (1, 3) and (5, 6).

Therefore, the answer is:

f is increasing on the intervals [0, 1) and (3, 5).

f is decreasing on the intervals (1, 3) and (5, 6).

(b)

From the given graph, we can see that f(x) has a local maximum at x = 2 and a local minimum at x = 4.

Therefore, the answer is: f has a local maximum at x = 2 and a local minimum at x = 4.

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The area of the 2 by 2 unit square is 4 square units.

Assuming that the shape being referred to is a square with sides of length 2 units as shown in the figure, the area can be calculated as follows:

Area of a square = (side length)^2

In this case, the side length is 2 units, so we can substitute this value into the formula:

Area = (2 units)^2 = 4 square units

Therefore, the area of the 2 by 2 unit square is 4 square units.

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The x-intercepts are (4,0)(5,0) of the following graph.

In the context of a graph, an x-intercept is a point where a curve or line crosses the x-axis. It is the point at which the value of y is zero. In other words, the x-intercept is the value of x when the function or equation crosses the x-axis.

To find the x-intercepts of a function, you can set y equal to zero and solve for x. The resulting values of x will be the x-intercepts.

In the context of a graph, a y-intercept is a point where a curve or line crosses the y-axis. It is the point at which the value of x is zero. In other words, the y-intercept is the value of y when the function or equation crosses the y-axis.

To find the y-intercept of a function, you can set x equal to zero and solve for y. The resulting value of y will be the y-intercept.

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The image coordinates of N'M'O' are (−9, −2), (−6, −1), and (−9, −5).

To translate the preimage 5 units to the left, we need to subtract 5 from the x-coordinates of all three vertices.

So the new coordinates of N', M', and O' will be:

N'((-4)-5, (-2)) = (−9, −2)

M'((-1)-5, (-1)) = (−6, −1)

O'((-4)-5, (-5)) = (−9, −5)

Therefore, the image coordinates of N'M'O' are (−9, −2), (−6, −1), and (−9, −5).

So the correct answer is N′(−9, −2), M′(−6, −1), O′(−9, −5).

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The true statement would be: The theoretical probability of landing on 11 is 12.5% while the experimental probability of landing on 11 is 30%

Experimental probability is a type of probability that is based on actual experimental results or data. It is the ratio of the number of times an event occurs to the total number of trials or experiments conducted.

For example, if you were to toss a coin 100 times and record the number of times it lands on heads, the experimental probability of the coin landing on heads would be the number of times it landed on heads divided by the total number of coin tosses (100 in this case). If the coin landed on heads 45 times out of 100, then the experimental probability of the coin landing on heads would be 45/100 or 0.45.

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Answer:

  a) sin(β) = 7/6 is false for any β

  b) angle β is not a real angle

Step-by-step explanation:

You want to use the Sine Law to show that a triangle with sides of lengths 6 and 10 and a smallest angle of 44.5° is impossible.

The Sine Law tells you side lengths are proportional to the sines of their opposite angles.

  sin(β)/10 = sin(44.5°)/6

  sin(β) = 10(0.7)/6

  sin(β) = 7/6

The angle β does not exist. The maximum value of the sine function is 1. There are no angles for which the sine is greater than 1.

The triangle is impossible because angle β cannot exist.

In effect, the angle 44.5° is too large, or side 10 is too long, or side 6 is too short for the figure to be a triangle.

__

Additional comment

In the realm of complex numbers, β ≈ π/2 -0.569618i radians, about (90 -32.64i)°. You cannot draw this complex angle with your real pencil.

a) For any, sin(β) = 1.17 is false.

b) Angle (β) is not a true angle.

In the branch of mathematics known as trigonometry, the relationships between the sides and angles of triangles are examined.

It involves the study of trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant that describe the connections between the sides and angles of a right triangle.

(a) Using the Sine Law, we have:

sin(β)/10 = sin(44.5°)/6

Multiplying both sides by 10, we get:

sin(β) = (10/6)sin(44.5°)

sin(β) = 1.17

However, this is impossible since the sine of an angle can never be greater than 1. Therefore, there is no value of β that satisfies the equation, and we cannot find the value of B.

(b) The fact that the sine of β is greater than 1 means that there is no angle whose sine is equal to that value.

Therefore, there is no possible angle for β that can satisfy the given conditions of the triangle.

This means that the triangle is impossible and cannot exist with the given side lengths and angle measures.

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Therefore, the common difference of the arithmetic sequence is 1.5.

To find the common difference of an arithmetic sequence, you need to know at least two consecutive terms of the sequence. The common difference is the constant amount by which each term increases or decreases from the previous term.

If you have the first term (a1) and the second term (a2) of an arithmetic sequence, you can find the common difference (d) using the formula:

[tex]d = a_2 - a_1[/tex]

For example, consider the arithmetic sequence: 3, 7, 11, 15, ...

Here, a1 = 3 and a2 = 7. Substituting these values in the formula, we get:

[tex]d = a_2 - a_1 = 7 - 3 = 4[/tex]

So, the common difference of this sequence is 4.

Let the first term of the arithmetic sequence be "a", and let the common difference be "d".

Then the seventh term would be:

[tex]a + 6d = 10.2[/tex]

And the twelfth term would be:

[tex]a + 11d = 17.7[/tex]

Now we have two equations with two variables. We can solve for "a" and "d" using these equations.

First, we can subtract the first equation from the second equation:

[tex](a + 11d) - (a + 6d) = 17.7 - 10.2[/tex]

Simplifying this equation, we get:

[tex]5d = 7.5[/tex]

Dividing both sides by 5, we get:

[tex]d = 1.5[/tex]

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The volume of the rectangular prism is 210cm³.

A rectangular prism is a three-dimensional solid shape that has six rectangular faces. Each face of the prism is a rectangle, and the opposite faces are congruent and parallel. The rectangular prism is also known as a rectangular cuboid or a rectangular parallelepiped. The rectangular prism has eight vertices (corners), twelve edges, and six faces. It is a special case of a parallelepiped, which is a three-dimensional solid with six parallelogram faces.

The surface area of the rectangular prism is:

Surface Area = 2(lw + lh +  wh)

Surface Area = 2(7cm x 6cm) + 2(7cm x 5cm) + 2(6cm x 5cm)

Surface Area = 84cm² + 70cm² + 60cm²

Surface Area = 214cm²

Hence, the rectangular prism has a 214 cm² surface area.

We must multiply the rectangular prism's length, width, and height in order to get its volume.

The volume of the rectangular prism is:

Volume = lwh

Volume = 7cm x 6cm x 5cm

Volume = 210cm³

Hence, the rectangular prism has a 210 cm³ volume.

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The complex fractions are simplified to;

(1/4 - 1/5)/(1/3)

To simplify this complex fraction, first simplify the numerator by finding a common denominator:

(1/4 - 1/5) = (5/20 - 4/20) = 1/20

Then, we can divide by the denominator by multiplying by its reciprocal:

1/20 ÷ (1/3) = 1/20 × (3/1) = 3/20

Therefore, the simplified complex fraction is 3/20.

(x-y)/(x⁻¹ - y⁻¹)

To simplify this complex fraction, use the formula for the difference of squares on the denominator:

x⁻¹ - y⁻¹ = (x - y)/(xy)

Then, we can rewrite the complex fraction as a division of two fractions:

(x-y)/(x⁻¹ - y⁻¹) = (x-y) × (xy)/(x - y) = xy

Therefore, the simplified complex fraction is xy.

((2x + 1)/(x- (1/x)) - (1/x))/(2x + (2x)/(1-1/x))

To simplify this complex fraction, first simplify each of the two fractions in the numerator and denominator separately:

(2x + 1)/(x- (1/x)) = [(2x² + x)/x]/[(x² - 1)/x] = (2x² + x)/(x² - 1)

(1/x) = (x⁻¹)

(2x + (2x)/(1-1/x)) = (2x + (2x)/(x-1))

= [(2x(x-1) + 2x)/ (x-1)] / [(x-1)/(x-1)]

= (2x² - 2x + 2x)/(x-1)

= (2x²)/(x-1)

Substituting these simplified fractions back into the original complex fraction:

[(2x² + x)/(x² - 1) - x⁻¹] ÷ [(2x²)/(x-1)]

= [(2x² + x)/(x² - 1) - (1/x)] × [(x-1)/(2x²)]

= [(2x² + x) - (x² - 1)]/[x(x² - 1)] × [(x-1)/(2x²)]

= (x² + x + 1)/[2x³(x-1)]

Therefore, the simplified complex fraction is (x² + x + 1)/[2x³(x-1)].

Find out more on complex fractions here: https://brainly.com/question/29015784

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Image transcribed:

60. (1/4 -1/5)/(1/3)

62. (x-y)/(x⁻¹ - y⁻¹)

64. ((2x 1)/(x- (1/x)) - (1/x))/(2x + (2x)/(1-1/x)

Answer:

  60. 3/20

  62.  -xy

  64.  (2x³ -x² +1)/(4x⁴ +2x³ -2x²)

Step-by-step explanation:

You want the simplified form of various complex fractions.

Fractions are added and divided in the usual ways:

  a +b/c = (ac +b)/c

  a/b +c/d = (ad +bc)/(bd)

  (a/b)/(c/d) = (ad)/(bc)

  (1/4 -1/5)/(1/3) = (5 -4)/(5·4)·(3/1) = 3/20

  (x -y)/(1/x -1/y) = (x -y)/((y -x)/(xy)) = xy(x -y)/(y -x) = -xy

  [tex]\dfrac{\dfrac{2x}{x-\dfrac{1}{x}}-\dfrac{1}{x}}{2x+\dfrac{2x}{1-\dfrac{1}{x}}}=\dfrac{\dfrac{2x^2}{x^2-1}-\dfrac{1}{x}}{2x+\dfrac{2x^2}{x-1}}=\dfrac{(2x^3-x^2+1)(x-1)}{x(x^2-1)(2x^2-2x+2x^2)}\\\\=\dfrac{2x^3-x^2+1}{2x^2(x+1)(2x-1)}=\boxed{\dfrac{2x^3-x^2+1}{4x^4+2x^3-2x^2}}[/tex]

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Additional comment

The factored form in 64 is useful for many purposes. "Simplified" usually means that factors are multiplied out. Note that we used (x²-1) = (x-1)(x+1).

If you were to describe the domain of the expression in 64, it would exclude x=1 as well as those values of x that make the remaining denominator factors zero.