Help!!! I don’t really understand this! :c

Answer:

10c

Step-by-step explanation:

When we have similar terms we can add or subtract them together.

Combine terms is like grouping similar terms together.

Accordingly, let us solve the given question.

5c + 5c

Combine the like terms to simplify the answer.

10c

Triangle PQR is translated to triangle P'Q'R'. (-3,3) and (-1,1) are the co-ordinates of Q' and R'.

The co-ordinates of Q' and R' when triangle PQR is translated to triangle P'Q'R' are (-3,3) and (-1,1), respectively.

A translation is a type of transformation where a shape is moved from one position to another.

However, the shape is not rotated, resized, or reflected in a translation.

Instead, it is moved in a straight line from one position to another.

A translation in a two-dimensional plane is specified by a vector, which specifies the distance and direction of the translation.

The co-ordinates of Q' and R' when triangle PQR is translated to triangle P'Q'R' are (-3,3) and (-1,1), respectively.

To determine these co-ordinates, we can use the following steps:

Determine the vector of the translation.

To do this, we find the difference between the corresponding vertices of the two triangles.

Here, we can take PQ as the base.

So, P'Q' = QRQR

= PQP' = P + (-1,2)

= (-2,3)Q'

= Q + (-1,2)

= (-3,3)R'

= R + (-1,2)

= (-1,1).

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According to the information of the graph the correct sentences are: D. Dilate ABC by a scale factor of 1/2 centered at the right angle, and E. Dilate ABC by a scale factor of 2 centered at the right angle.

To show that two triangles are similar, we need to demonstrate that they have the same shape but not necessarily the same size. This can be done using similarity transformations such as dilation, rotation, and reflection. Statements A, B, and C involve rotations which would change the shape of the triangle, therefore they do not demonstrate similarity.

On the other hand, statement D suggests dilating triangle ABC by a scale factor of 1/2 centered at the right angle. This transformation will change the size of the triangle but maintain the same shape, which is a property of similar triangles. Similarly, statement E suggests dilating triangle ABC by a scale factor of 2 centered at the right angle, which again maintains the same shape but changes the size. Therefore, statements D and E demonstrate that triangle ABC and triangle LMN are similar.

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Answer: 7, and it occurs when x = 5 and y = 2

Step-by-step explanation:

1/5 x + 1/3 y = 1 (Equation 1)

x + y = ? (Equation 2)

We want to find the maximum value of x + y, so we need to maximize each of the variables x and y subject to the constraint given in Equation 1.

To solve for one of the variables, we can isolate it in terms of the other variable. Let's solve for y:

1/5 x + 1/3 y = 1

1/3 y = 1 - 1/5 x

y = 3(1 - 1/5 x)

Now we can substitute this expression for y into Equation 2:

x + y = x + 3(1 - 1/5 x)

Expanding the expression, we get:

x + y = x + 3 - 3/5 x

x + y = 3 + 2/5 x

So we can see that x + y is a linear function of x with slope 2/5 and y-intercept 3. To maximize x + y, we want to maximize x. However, x is a non-negative integer, so the maximum value of x occurs when x = 5 (since any larger value would result in a fraction in the expression for y).

Therefore, the maximum value of x + y is:

x + y = 3 + 2/5 x

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

x + y = 5 + 2

x + y = 7

So the maximum value of the sum of these two non-negative integers is 7, and it occurs when x = 5 and y = 2.

Answer:

[tex]A = 212.1 \text{ cm}^2[/tex]

Step-by-step explanation:

We can identify the bases of the trapezoid as:

9cm and 21.3 cm

(notice how they are parallel)

We know the height is 14 cm.

Using these values, we can solve for the trapezoid's area.

[tex]A = \dfrac{1}{2}(b_1 + b_2) \cdot h[/tex]

↓ plugging in the values

[tex]A = \dfrac{1}{2}(9 + 21.3) \cdot 14[/tex]

↓ simplifying the parentheses

[tex]A = \dfrac{1}{2}(30.3) \cdot 14[/tex]

↓ multiplying 14 and (1/2)

[tex]A = 7 \cdot 30.3[/tex]

↓ multiplying

[tex]A = 212.1 \text{ cm}^2[/tex]

Answer:

x ≈ 51°

Step-by-step explanation:

Use trigonometry:

[tex] \sin(x°) = \frac{7}{9} [/tex]

Use the reverse (SHIFT, sin (7/9)) function on your calculator to find x:

x ≈ 51°

The answer of the given question based on compass and Straight edge construction is , option (b) False.

A pentagon is a geometric shape that has five straight sides and five angles. It is a polygon, which means it is a closed shape made up of straight line segments. The word "pentagon" come from Greek word "penta" meaning "five" and "gonia" meaning "angle".

There are several types of pentagons, including regular and irregular pentagons. A regular pentagon has five equal sides and angles, while an irregular pentagon has sides and angles of different lengths and measures.

False.

The Greeks believed that all constructions could be made with only a compass and straightedge, but some constructions were more difficult than others. In fact, they were able to make remarkable geometric constructions using only these two tools, such as trisecting an angle, duplicating a cube, and constructing a regular pentagon. However, in the 19th century, the discovery of new mathematical methods showed that some constructions, like squaring the circle, were actually impossible to complete with just a compass and straightedge.

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A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

Inequality refers to the state of being unequal, or not having equal access to resources, opportunities, or privileges. Inequality can manifest in many different ways, including but not limited to income inequality, wealth inequality, social inequality, gender inequality, racial inequality, and educational inequality.

Inequality can be caused by a variety of factors, including systemic discrimination, unequal distribution of resources, and societal structures that favor certain groups over others. It can have a significant impact on individuals and communities, leading to disparities in health, education, economic opportunities, and overall well-being. Addressing inequality requires a multifaceted approach that involves policies and initiatives aimed at promoting equal access to resources, opportunities, and rights for all individuals.

Which number line represents the solution set for the inequality 3(8 – 4x) < 6(x – 5)?

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the left.

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the left.

A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the right.

To solve the inequality3(8 – 4x) < 6(x – 5)[tex]3(8 - 4x) < 6(x – 5)[/tex], we first simplify both sides of the inequality:

[tex]3(8-4x) < 6(x -5)[/tex]

[tex]24 - 12x < 6x - 30[/tex]

[tex]54 < 18x[/tex]

[tex]3 < x[/tex]

So, the solution set for the inequality is x > 3. We represent this on a number line by placing an open circle at 3 and drawing a bold line pointing to the right to indicate all the values of x that satisfy the inequality. Therefore, the correct answer is:

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

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Answer:

it's 3

Step-by-step explanation:

The derivative of the sigmoid function is f(x) × (1 - f(x)).Answer: f(x) × (1 - f(x)).

As per the question, we are supposed to determine the derivative of the sigmoid function.

The sigmoid function is given as,

[tex]f(x) = 1 / (1 + e^-x)[/tex]

To determine the derivative of the sigmoid function, we first find out the derivative of f(x) with respect to x using the quotient rule,

[tex]f'(x) = d/dx [ 1 / (1 + e^-x) ][/tex]

[tex]f'(x) = [ (d/dx)[1] × (1 + e^-x) - 1 × (d/dx)[1 + e^-x] ] / [ (1 + e^-x)^2 ][/tex]

[tex]f'(x) = [ 0 × (1 + e^-x) - (-e^-x) ] / [ (1 + e^-x)^2 ][/tex]

[tex]f'(x) = e^-x / (1 + e^-x)^2[/tex]

Now we are supposed to write the answer in terms of f(x),

[tex]f'(x) = e^-x / (1 + e^-x)^2f(x) = 1 / (1 + e^-x)[/tex]

Substituting f(x) in the above equation,

f'(x) = f(x) × (1 - f(x))

Therefore, the derivative of the sigmoid function is f(x) × (1 - f(x)).Answer: f(x) × (1 - f(x)).

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Answer:

a = 3
b = 8
c = 4

Step-by-step explanation:

We can identify the coefficients a, b, and c by comparing the equation with the standard form of a quadratic equation:

ax^2 + bx + c = 0

(always use this for these kinds of problems)

So, in the given equation:

The coefficient of x^2 is a = 3

The coefficient of x is b = 8

The constant term is c = 4

To better understand your question, it is important to identify the key terms and concepts that are being used.

Title: Understanding Terms in a Question

1. . Look for specific words or phrases that relate to the topic at hand, and try to understand their meaning within the context of the question.

2. It can also be helpful to break the question down into smaller parts or sub-questions, and focus on each one individually.

3. If you are unsure about the meaning of a particular term, do some research or consult a dictionary or other reference source.

4. By taking the time to understand the terms and concepts in a question, you can improve your ability to answer it accurately and effectively.

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The two-column proofs that line segments AC ≅ EC are shown below

Given that

AB || DE

BC ≅ DC

The proof is as follows

Statement                                                Reason

AB || DE and BC ≅ DC                            Given

AB ≅ DE                                                   CPCTC

AC ≅ EC                                                   CPCTC (proved)

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Therefore, f(11) can either be -16 or 20, depending on whether we use the recursive or explicit rule.

In mathematics, a graph is a visual representation of the relationship between two or more variables. It is a collection of points (or vertices) that are connected by lines or curves (called edges) to show the relationships between them. Graphs are commonly used in many different fields of study, including mathematics, science, engineering, and social sciences, to represent and analyze data, identify patterns, and make predictions.

Here,

We can observe that this is an arithmetic sequence, where each term is obtained by adding a constant difference to the previous term. To find the recursive rule, we can use the formula:

f(n) = f(n-1) + d

where f(n-1) is the previous term, d is the common difference, and f(n) is the nth term. Using the given values, we can find that the common difference is d = -6. So the recursive rule for the sequence is:

f(1) = 80

f(n) = f(n-1) - 6

To find the explicit rule, we can use the formula:

f(n) = a + (n-1)d

where a is the first term, d is the common difference, and n is the term number.

Using the first two terms, we can find that:

a = 80

d = -6

So the explicit rule for the sequence is:

f(n) = 80 - 6(n-1)

To find f(11), we can substitute n = 11 into either the recursive or explicit rule:

Using the recursive rule:

f(11) = f(10) - 6

= (f(9) - 6) - 6

= (f(8) - 12) - 6

= (f(7) - 18) - 6

= (f(6) - 24) - 6

= (f(5) - 30) - 6

= (20 - 30) - 6

= -16

Using the explicit rule:

f(11) = 80 - 6(11-1)

= 80 - 6(10)

= 20

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In light of this, it will take roughly 2.25 years for the populace to reach 3050,that shows growth rate. The equation to calculate the number of fishes in a pond after t years would be equal to t = -ln (0.12125) / r

The logistic development model can be used to calculate an equation for the growth rate of fish P(t) after t years:

Where[tex]: P(t) = K / (1 plus A e(-rt))[/tex]

[tex]r[/tex]= growth rate, where[tex]K =[/tex] carrying capacity [tex]= 6100 A =[/tex] starting population[tex]= 500[/tex]

T equals time in years

Using the given data, we can determine the increase rate:

In the first year, the quantity of fish increased from[tex]500 to 740:[/tex]

[tex]740 = 6100 / (1 + A e^(-r1))[/tex]

We arrive at [tex]A e(-r*1) = 8.24324[/tex] after solving for [tex]A e(-r).[/tex]

Given that P(t) must lie half way between A and K in order for there to be a maximal growth rate, the carrying capacity is [tex]6100:[/tex]

[tex]3050 = (6100 + 500) / 2 = 3300[/tex]

As a result, if [tex]P(t) = 3300,[/tex] we have:

[tex]3300 = 6100 / (1 + A e^(-rt))[/tex]

We arrive at [tex]A e(-rt) = 1.84848[/tex] after solving for [tex]A e(-rt).[/tex]

The following is the result of substituting the numbers of [tex]A e(-r*1)[/tex] and [tex]A e(-rt)[/tex] into the equation for P(t):

We can put into an expression and solve for t to determine how long it will take for the population to reach [tex]3050:[/tex]

[tex]3050 = 6100 / (1 + 8.24324 e^(-rt))[/tex]

[tex]e(-rt) = 0.12125 -rt = ln = 1 + 8.24324 e(-rt) = 2(0.12125)[/tex]

[tex]t = -ln(0.12125) / r[/tex]

Although we are unsure of r's precise value, we do know that [tex]A e(-rt) = 1.84848.[/tex]

at[tex]t = 3300, P(t).[/tex] When we enter these numbers into the solution, we obtain:

[tex]1.84848 = A e^(-r1)[/tex]

[tex]A = 1.84848 / e^(-r1)[/tex]

We obtain the following equation for t by substituting this value of A:[tex]t = -ln(0.12125) / r t = -ln(0.12125) / ln(1.84848 / e(-r*1))[/tex]

The precise value of r is not required to provide an answer, but we can solve for it using a numerical approach like trial and error or a graphing method.

The logistic development model can be used to calculate an equation for the quantity of fish P(t) after t years:

Where: [tex]P(t) = K / (1 plus A e(-rt))[/tex]

r = growth rate, where K = carrying capacity =[tex]6100 A[/tex] = starting population [tex]= 500[/tex]

T equals time in years

Using the given data, we can determine the increase rate:

In the first year, the quantity of fish increased from 500 to 740:

[tex]740 = 6100 / (1 + A e^(-r1))[/tex]

We arrive at A e(-r*1) = 8.24324 after solving for A e(-r).

Given that P(t) must lie half way between A and K in order for there to be a maximal growth rate, the carrying capacity is 6100:

[tex]3050 = (6100 + 500) / 2 = 3300[/tex]

As a result, if P(t) = 3300, we have:

[tex]3300 = 6100 / (1 + A e^(-rt))[/tex]

We arrive at c[tex]A e(-rt) = 1.84848[/tex] after solving for [tex]A e(-rt).[/tex]

The following is the result of substituting the numbers of A e(-r*1) and A e(-rt) into the equation for P(t):

[tex]P(t) = 6100 / (1 + 8.24324 e^(-rt))[/tex]

We can put P(t) = 3050 into an expression and solve for t to determine how long it will take for the population to reach 3050:

[tex]3050 = 6100 / (1 + 8.24324 e^(-rt))[/tex]

[tex]e(-rt) = 0.12125 -rt = ln = 1 + 8.24324 e(-rt) = 2(0.12125)[/tex]

[tex]t = -ln(0.12125) / r[/tex]

Although we are unsure of r's precise value, we do know that[tex]A e(-rt) = 1.84848.[/tex]

a[tex]t t = 3300, P(t).[/tex] When we enter these numbers into the solution, we obtain:

[tex]1.84848 = A e^(-r1)[/tex]

[tex]A = 1.84848 / e^(-r1)[/tex]

We obtain the following equation for t by substituting this value of A:[tex]t = -ln(0.12125) / r t = -ln(0.12125) / ln(1.84848 / e(-r*1))[/tex]

Assuming r = 0.1 as a sample growth rate, we obtain[tex]t = -ln (0.12125) / ln (1.84848 / e(-0.1*1))[/tex]

The precise value of r is not required to provide an answer, but we can solve for it using a numerical approach like trial and error or a graphing calculator. By assuming a growth rate, we can calculate how long it will take for the populace to reach 3050. Assuming r = 0.1 as a sample growth rate, we obtain[tex]t = -ln(0.12125) / ln(1.84848 / e(-0.1*1))[/tex]

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Answer: i would say $1.60 because 20% of $8.00 is $1.60

Each friend earned an amount of $1.6.

Percentage is defined as the parts of a number per fraction of 100. We have to divide a number with it's whole and then multiply with 100 to calculate the percentage of any number.

So the percentage actually means a part per 100.

Percentage is usually denoted by the symbol '%'.

Given that,

A group of friends worked together at a lemonade stand.

They earned $8.00 in all.

Each friend was paid 20% of the total earnings.

Total earnings = $8

Each friend earn = 20% × 8

                            = (20/100) × 8

                            = 0.2 × 8

                            = 1.6

Hence the amount each friend earned is $1.6.

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Step-by-step explanation:

 For a RIGHT triangle , remember  S-O-H-C-A-H-T-O-A ?

    Cos (Φ) = Adjacent leg / Hypotenuse

  so   cos D =    sqrt (78) / DF =  sqrt (78) / 28

Rounding to the nearest tenth, the measure of UV is 7.1.

An angle is a measure of space between two lines or planes. It is measured in degrees, using a protractor, and can range from 0 to 360. Angles can be either acute, obtuse, right, straight, reflex, or full. Acute angles are angles that measure less than 90 degrees, obtuse angles measure more than 90 degrees, right angles measure exactly 90 degrees, straight angles measure 180 degrees, reflex angles measure more than 180 degrees, and full angles measure 360 degrees.

To find the measure of UV in ∆ UVW, we will use the law of cosines. The law of cosines states that [tex]c^{2} =a^{2} +b^{2} -2ab cos C[/tex], where c is the length of the side opposite angle C, a and b are the lengths of the other two sides, and C is the angle opposite side c. In this case, side c is UV, a is UW, b is VW, and C is ∠ W.

Plugging in the given values, we get:

[tex]U^{2}V^{2}[/tex] = [tex]9^{2} + 12^{2} - 2 *9* 12 cos *35[/tex]

[tex]U^{2} V^{2}[/tex] = 81 + 83.44 – 162.08 cos 35°

[tex]U^{2} V^{2}[/tex] = 164.44 – 114.45

[tex]U^{2} V^{2}[/tex] = 49.99

Taking the square root of both sides, we get:

UV = 7.086

Rounding to the nearest tenth, the measure of UV is 7.1.

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can you add the question?

Add a picture of the question.

The common end point I believe :)

The surface area of the given triangular prism is approximately 58.26 square units.

what is triangular prism?

A triangular prism is a three-dimensional geometric shape that consists of two congruent parallel triangles as its bases and three rectangular faces that connect the corresponding sides of the two triangles.

In the given question,

To find the surface area of a triangular prism, we need to add the area of all the faces of the prism. A triangular prism has two congruent triangular bases and three rectangular lateral faces.

First, let's calculate the area of the triangular bases. We can use the formula for the area of a triangle:

Area of a triangle = (1/2) * base * height

In this case, the base of the triangle is 5 and the height is 6. To find the height, we need to use the Pythagorean theorem because we are given the slant height of the triangular prism, which is the hypotenuse of the right triangle formed by the base, the height, and the slant height. The Pythagorean theorem is:

slant height² = base² + height²

Substituting the given values, we get:

6.5² = 5² + height²

42.25 = 25 + height²

height² = 17.25

height = √17.25 ≈ 4.15

Now we can calculate the area of each triangular base:

Area of a triangular base = (1/2) * base * height

Area of each triangular base = (1/2) * 5 * 4.15

Area of each triangular base ≈ 10.38

Next, we need to calculate the area of each rectangular lateral face. The length of each lateral face is the same as the base of the triangle, which is 5, and the width is the same as the width of the prism, which is 2.5. The height of each lateral face is the same as the height of the prism, which is 6. Therefore, the area of each rectangular lateral face is:

Area of a rectangular lateral face = length * width

Area of each rectangular lateral face = 5 * 2.5

Area of each rectangular lateral face = 12.5

Finally, we can find the total surface area of the prism by adding up the areas of the two triangular bases and the three rectangular lateral faces:

Total surface area = 2 * area of a triangular base + 3 * area of a rectangular lateral face

Total surface area ≈ 20.76 + 37.5

Total surface area ≈ 58.26

Therefore, the surface area of the given triangular prism is approximately 58.26 square units.

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Options A and B are both correct. Options C, D, and E are not correct, as they do not take into account the discount that James is receiving.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. Percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

The correct calculations to determine how much James will pay in total are:

First, we need to calculate how much 10% of $38.50 is:

10% of $38.50 = 0.1 × $38.50 = $3.85

So, James will receive a discount of $3.85.

To determine how much James will pay in total, we can use either of the following calculations:

A. James' total cost after the 10% discount would be: 38.5 - 0.9(38.5) = 38.5 - 34.65 = $3.85

B. The 10% discount on the total cost can be calculated as 0.1(38.5) = $3.85, which can be subtracted from the total cost to get James' final cost as 38.5 - 3.85 = $34.65

C. James' final cost after the 10% discount would be 1.1(38.5) = $42.35, which is not the correct answer as it does not take into account the discount.

D. James' total cost after the 10% discount would be: 0.9(38.5) = $34.65, which is the correct answer.

E. The calculation 0.1(38.5) = $3.85 represents the amount of the discount, and is not the final cost.

Therefore, options A and B are both correct. Options C, D, and E are not correct, as they do not take into account the discount that James is receiving.

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1) Given that 1789 children visited a Taco Bell in 24 hours. We can calculate the number of students who went every hour by dividing this number by the number of hours:

[tex]1789 / 24 = 74.54\\\frac{1789}{24} = 74.54[/tex]

We get 75 students each hour, rounded up to the nearest whole number.

If 24 children shared, we can calculate the total number of children who shared by multiplying the number of hours by the number of children shared every hour:

24 x 75 = 1800

As a result, 1800 children shared.

2)If 1638 men cheated on their wives in one week, multiplying by 7 gives us the number of men who cheated on their wives in seven weeks:

1638 x 7 = 11,466

As a result, 11,466 men cheated on their wives in just 7 weeks.

c) If 47% of the men who cheated on their spouses in 7 weeks cheated numerous times before, we can calculate the number of men who cheated previously by multiplying the total number of men who cheated by the proportion of men who cheated previously:

11,466 x 0.47 = 5,390.02

Rounding to the nearest whole number, we get 5,390 men who have cheated in the past.

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Answer: 4/5

Step-by-step explanation:

Five equal sides = 5
4 is just one side, so = 1
Not landing in 4 = 4/5

Answer: 4/5 or 80%

Step-by-step explanation:

Since the spinner has five equal sections, each numbered 1 through 5, the probability of landing on any particular number is 1/5.

To find the probability of not landing on 4, we need to add up the probabilities of landing on all the other numbers and subtract that sum from 1 (since the sum of all possible outcomes must equal 1).

The probability of landing on 4 is 1/5, so the probability of not landing on 4 is:

1 - 1/5 = 4/5

Therefore, the probability of not landing on 4 is 4/5 or 0.8, which is equivalent to an 80% chance.

Hope this helped!

The parametric equation of the line is given by;  x(t)=-2 + 5ty(t)=2 - 3tz(t)=5 - 3t

To find the vector and parametric equations for the line through the point P(-2, 2, 5) and orthogonal to the plane 5x - 3y - 3z = -3, the following  

Write the equation of the plane.5x - 3y - 3z = -3  is the given equation

Find a vector that is orthogonal to the planeThis can be done by finding the normal vector of the plane. Using the coefficients of x, y, and z in the equation of the plane.  A normal vector of the plane is obtained and simplified:<5,-3,-3>.

Find the vector equation of the line.

Let L be the line through P(-2, 2, 5) and orthogonal to the plane. A direction vector of the line,  v is the normal vector of the plane (which is <5,-3,-3>), a point P(-2, 2, 5) on the line can be used to obtain the vector equation of the line:

Parametric form (parameter t, and passing through P when t = 0):x(t)=-2 + 5ty(t)=2 - 3tz(t)=5 - 3t

The vector form:<-2, 2, 5> + t<5,-3,-3> The parametric equation of the line is given by;  x(t)=-2 + 5ty(t)=2 - 3tz(t)=5 - 3t

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Unauthorized use of celebrity name and misleading statements are illegal to include in a sales letter.

A sales letter comprises a format that follows a certain set of instruction that should always be implemented to form an ideal sales letter, this set of instruction are as follows

Furthermore, a sales letter provides a basic layout of the needs of the writer and how they can help in providing aid for the needs of the reader.

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The solution set can be written in parametric vector form as: [tex]X = x3 [-1 0 1]^T + x2 [-2 1 0]^T[/tex]

We can rewrite the system of equations in matrix form as AX = 0, where [tex]A =[2 2 4]\\[-4 -4 -8]\\[0 -3 -9][/tex]

[tex]X = [x1 x2 x3]^T[/tex]

To solve for the solution set, we can row reduce the augmented matrix [tex][A|0].\\[2 2 4|0]\\[-4 -4 -8|0]\\[0 -3 -9|0][/tex]

[tex][2 2 4|0]\\[0 0 0|0]\\[0 -3 -9|0][/tex]

[tex][2 2 4|0]\\[0 0 0|0]\\[0 1 3|0][/tex]

[tex][2 2 4|0]\\[0 0 0|0]\\[0 1 3|0][/tex]

[tex][1 1 2|0]\\[0 0 0|0]\\[0 1 3|0][/tex]

Therefore, the system has two free variables, x2 and x3, while x1 is a pivot variable. where x2 and x3 are arbitrary constants.

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We have the following system of equations:

x₁ + 0.5x₂ + 0.1x₃ = 100

0.25x₁ + x₂ + 0.2x₃ = 75

0.3x₁ + 0.4x₂ + x₃ = 200

Statement gives the equality of different mathematical expressions is called as an equation.

Let x₁, x₂, and x₃ be the outputs of I₁, I₂, and I₃, respectively. Then the total production must satisfy the following system of equations:

For I₁:

For I₂:

For I3:

Therefore, the system equations are:

x₁ + 0.5x₂ + 0.1x₃ = 100

0.25x₁ + x₂ + 0.2x₃ = 75

0.3x₁ + 0.4x₂ + x₃ = 200

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