In this part, you will prove that7k+1−1is divisible by 6 . By inductive hypothesis, since 6 evenly divides integermsuch that=6 m. Hence,7k=It follows that,7k+1−1=7Sincemis an integer and integers are closed under , there exists an Sincemis an integer a must be an integer. Therefore,7k+1−1is divisible by 6 .

In Problem 4, there are (i) 495 different possible orders for the group when they collectively order four different dishes to share, and (ii) 20,736 different possible orders for the group when each individual orders a main course.

(i) To find the number of ways to order four different dishes out of 12, we use combinations. This is calculated as C(12,4) = 12! / (4! * (12-4)!), which equals 495 possible orders.

(ii) Since there are 12 dishes and each of the four friends can choose any dish, we use permutations. The number of possible orders is 12⁴, which equals 20,736 different orders.

For passwords, there are (i) 306,380,448 passwords of length 7 with at least one digit, and (ii) 282,475,249 passwords of length 7 with at least one digit and one letter.

(i) There are 10 digits and 26 lowercase letters. Total possibilities are (10+26)⁷. Subtract the number of all-letter passwords: 26^7. Result is (36⁷) - (26⁷) = 306,380,448.

(ii) Subtract the number of all-digit passwords from the previous result: 306,380,448 - (10⁷) = 282,475,249 different passwords.

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The linear relationship between x = year and y = profit is y = 4x + 2.

The company's profit at the end of 2 years is $10 million.

The company's profit at the end of 5 years is $22 million.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (14 - 6)/(3 - 1)

Slope (m) = 8/2

Slope (m) = 4

At data point (1, 6) and a slope of 4, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 6 = 4(x - 1)  

y = 4x - 4 + 6

y = 4x + 2

When x = 2 years, the profit is given by;

y = 4(2) + 2 = $10 million

When x = 5 years, the profit is given by;

y = 4(5) + 2 = $22 million.

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The calculated value of x in the triangle is 4.79 feet

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

X

7 feet

43.2°

The value of x in the triangle can be calcuated using the following sine rule

sin(43.2) = x/7

Make x the subject of the above equation

So, we have

x = 7 * sin(43.2)

Evaluate the products

x = 4.79

Hence, the value of x is 4.79 feet

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The equation shows that the height of the stack is directly proportional to the number of cups in the stack, with a proportionality constant of 2 cm.

The stack of 8 cups is 16 cm tall.

To determine the height of 1 cup, we can divide the height of the stack (16 cm) by the number of cups (8):

1 cup = 16 cm ÷ 8 cups = 2 cm

The general relationship between the height of the stack (h) and the number of cups in the stack (n) can be expressed as:

h = n × 2 cm

Thus, this equation shows that the height of the stack is directly proportional to the number of cups in the stack, with a proportionality constant of 2 cm.

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1)The net change in the value of the function f(x) = 6x - 5 between the given inputs 1 and 6 is 30.

2)The net change in the value of the function g(t) = 1 - t² between the given inputs -4 and 9 is -65.

1. To find the net change in the value of the function f(x) = 6x - 5 between the given inputs 1 and 6:

Follow these steps:

Step 1: Calculate f(1)
f(1) = 6(1) - 5 = 6 - 5 = 1

Step 2: Calculate f(6)
f(6) = 6(6) - 5 = 36 - 5 = 31

Step 3: Find the net change
Net change = f(6) - f(1) = 31 - 1 = 30

The net change in the value of the function f(x) = 6x - 5 between the given inputs 1 and 6 is 30.

2. To find the net change in the value of the function g(t) = 1 - t² between the given inputs -4 and 9:

Follow these steps:

Step 1: Calculate g(-4)
g(-4) = 1 - (-4)² = 1 - 16 = -15

Step 2: Calculate g(9)
g(9) = 1 - 9² = 1 - 81 = -80

Step 3: Find the net change
Net change = g(9) - g(-4) = -80 - (-15) = -80 + 15 = -65

The net change in the value of the function g(t) = 1 - t² between the given inputs -4 and 9 is -65.

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The given differential equation is an Euler equation, the general solution is y = c1 + c2/[tex]x^4[/tex] and the singular point of the differential equation is x = 0

You are correct, this is an Euler equation. To solve it, we can make the substitution y = [tex]x^r[/tex]. Then we have:

Substituting these into the original equation, we get:

x²(r(r-1)[tex]x^(^r^-^2^)[/tex]) + 5x(r[tex]x^(^r^-^2^)[/tex]) + 4[tex]x^r[/tex]= 0

Simplifying, we have:

r(r+4)[tex]x^r[/tex] = 0

Since [tex]x^r[/tex] is never zero, we must have r(r+4) = 0. This gives us two possible values for r: r = 0 and r = -4.

Thus, the general solution is:

y = c1 + c2/[tex]x^4[/tex]

This solution is valid in any interval not including the singular point x = 0, which is the singular point of the differential equation.

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the probability that 1.20 ≤ z ≤ 1.85 is approximately 0.0822.To find the probability that 1.20 ≤ z ≤ 1.85, we need to use the standard normal distribution table or calculator.

First, we find the area to the left of 1.85 in the standard normal distribution table, which is 0.9671. Then, we find the area to the left of 1.20 in the standard normal distribution table, which is 0.8849.

To find the probability that 1.20 ≤ z ≤ 1.85, we subtract the area to the left of 1.20 from the area to the left of 1.85:

0.9671 - 0.8849 = 0.0822

Therefore, the probability that 1.20 ≤ z ≤ 1.85 is approximately 0.0822.

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Answer:

Number 3 and 4 are correct, but I have no clue about 1 or 2.

Step-by-step explanation:

I'm just gonna start with number 4

if you put 7/2 into decimals you get 3.5    7/2 is greater than 3

number 3.   -4 is in the negative zone, so it is less than 15 which is positive

if I were you, I would guess that number 1 is false. but i cant be sure

The joint pdf of y1, y2, and y3 is f(y1, y2, y3) = 2[tex]e^(^-^y^1^-^y^3^)[/tex](y1y3)⁻². They are not mutually independent, as their joint pdf cannot be factored into individual pdfs of y1, y2, and y3.

To find the joint pdf, first note the transformations: x1 = y3/y1, x2 = y3/y2, and x3 = y1y2y3. The Jacobian of this transformation is |J| = |(∂(x1, x2, x3)/∂(y1, y2, y3))| = |2y1y2y3²|.

Next, find the joint pdf of x1, x2, and x3: f(x1, x2, x3) = [tex]e^-^x^1e^-^x^2e^-^x^3[/tex] , since they are i.i.d. with exp(1) distribution. Now, apply the transformation and Jacobian: f(y1, y2, y3) = f(x1, x2, x3)|J| =  [tex]e^-^x^1e^-^x^2e^-^x^3[/tex] (2y1y2y3²) = 2[tex]e^(^-^y^1^-^y^3^)[/tex](y1y3)⁻². As the joint pdf cannot be factored into individual pdfs of y1, y2, and y3, they are not mutually independent.

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Answer:      7≤x≤9

Step-by-step explanation:

Explanation is in image but i forgot to write final answer form.  It's up top.

The exact values using law of negative exponents and reciprocals are:

a) 4⁻¹ = 1/4

b) 2⁻³ = 1/8

c) 3⁻⁴ = 1/81

The law of negative exponents and reciprocals states that:

Any non-zero number that is raised to a negative power will be equal to its reciprocal raised to the opposite positive power. This means that, an expression raised to a negative exponent will be equal to 1 divided by the expression with the sign of the exponent changed.

a) The number is given as: 4⁻¹

Applying the law of negative exponents and reciprocals, we have:

4⁻¹ = 1/4¹

= 1/4

b) The number is given as: 2⁻³

Applying the law of negative exponents and reciprocals, we have:

2⁻³ = 1/2³

= 1/8

c) The number is given as: 3⁻⁴

Applying the law of negative exponents and reciprocals, we have:

3⁻⁴ = 1/3⁴

= 1/81

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Correct question is:

Find the exact values of the following, giving your answers as fractions

a) 4⁻¹

b) 2⁻³

c) 3⁻⁴

Hi! The probability that Chris will win the election is 5/8 or 0.625, and the probability that Chris does not win is 3/8 or 0.375. It has been mentioned that the odds in favor of Chris winning the election are 5:3.

a) To find the probability that Chris wins, we can use the formula:
Probability = (Odds in favor) / (Odds in favor + Odds against)
In this case, the odds in favor are 5, and the odds against are 3. So, the probability of Chris winning is:
Probability = 5 / (5 + 3)
Probability = 5 / 8
The probability that Chris wins the election is 5/8 or 0.625.

b) To find the probability that Chris does not win, we can simply subtract the probability of Chris winning from 1:
Probability (Chris does not win) = 1 - Probability (Chris wins)
Probability (Chris does not win) = 1 - 5/8
Probability (Chris does not win) = 3/8 or 0.375

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Hi! The probability that Chris will win the election is 5/8 or 0.625, and the probability that Chris does not win is 3/8 or 0.375. It has been mentioned that the odds in favor of Chris winning the election are 5:3.

a) To find the probability that Chris wins, we can use the formula:
Probability = (Odds in favor) / (Odds in favor + Odds against)
In this case, the odds in favor are 5, and the odds against are 3. So, the probability of Chris winning is:
Probability = 5 / (5 + 3)
Probability = 5 / 8
The probability that Chris wins the election is 5/8 or 0.625.

b) To find the probability that Chris does not win, we can simply subtract the probability of Chris winning from 1:
Probability (Chris does not win) = 1 - Probability (Chris wins)
Probability (Chris does not win) = 1 - 5/8
Probability (Chris does not win) = 3/8 or 0.375

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Complete compensation= 832 individual items x Rs10/unit coming out to be: Rs. 8320

To determine the aggregate salary of a laborer, we will begin by computing the whole quantity of units manufactured per calendar month and then increase it by the wage rate for each unit.

Total units produced every month:

Monthly business days = 26

Productivity every hour = 4 individual items

Quantity of daily working hours = 8 hours

Units generated in one day = Productivity every hour multiplied by the Quantity of daily working hours

Units generated in one day are equal to 4 units/hour x 8 hours/day totalling= 32 individual items/day

Whole number units made each month = Units produced every day multiplied Monthly occupation days

Entire units produced each calendar month are equivalent to 32 individual items/day x 26 days which equals= 832 individual items/month.

The wage rate obtained receives Rs.10/individual item

Full pay gained is ascertained using Total units produced every month multiplied Wage rate Ruppees/Rs.10 for every unit.

Complete compensation= 832 individual items x Rs10/unit coming out to be: Rs. 8320

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The open interval where the function is increasing at  (-∞, ∞), decreasing at 0, or constant at 3.

Here we have the graph and through the graph we have to find the open interval(s) where the following function is increasing, decreasing, or constant.

While we looking into the give  graph we have identified that the function of the graph is determined as.

=> y = 3x + 2

To determine whether the function y = 3x + 2 is increasing, decreasing, or constant, we can analyze its first derivative.

The first derivative of y = 3x + 2 is y' = 3.

As the first derivative is a constant (y' = 3), the original function is continuously increasing for all values of x, and there are no intervals in which it is decreasing or constant.

Thus, the open interval where y = 3x + 2 is increasing is (-∞, ∞).

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In linear algebra, the determinant is a scalar value that can be computed from a square matrix.

To find the values of k for which the system has a nontrivial solution, we need to first analyze the given system of linear equations:

x1 + kx2 = 0
kx1 + 9x2 = 0

A nontrivial solution means there exists a solution where x1 and x2 are not both equal to zero. We can find such solutions by finding the determinant of the coefficients matrix and setting it equal to zero:

| 1   k |
| k   9 |

The determinant is calculated as follows:

Determinant = (1 * 9) - (k * k) = 9 - k^2

For a nontrivial solution, the determinant must be equal to zero:

9 - k^2 = 0

Now, solve for k:

k^2 = 9
k = ±3

So the values of k for which the system has a nontrivial solution are k = -3 and k = 3. Your answer: -3, 3

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For the term 1/(s²+4)(s²+1), the partial fraction decomposition would be A/(s²+4) + B/(s²+1), where A and B are constants that can be solved using algebraic equations.

The Laplace transform of e^(-2πs)sin(t-2π) is (s/(s²+1)² + 4π/(s²+1)). For the term (3s+6)/(s²+4), you can separate it into 3s/(s²+4) and 6/(s²+4), and their Laplace transforms would be (3/2)cos(2t) and (3/2)sin(2t), respectively. Once you have the Laplace transforms for each term, you can use linearity of Laplace transforms to get the solution of the given initial value problem.

Laplace transforms are a mathematical tool used to transform a function of time into a function of complex frequency. This transformation allows for the solving of differential equations, particularly those with initial conditions, by converting them into algebraic equations that can be easily solved.

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Answer:

$1.720.34

Step-by-step explanation:

to do this problem we can use the exponential growth formula A=p(1+r)^t

substituting our values we get

A = 1,557.00(1+0.02)^5

after solving the equation for A we get that

A after 5 years will be $1,720.34

The exact value of the expression cos(-10pi/3) is -1/2.

To find the exact value of cos(-10pi/3), follow these steps:

1. Determine the equivalent positive angle: Since the cosine function has a period of 2pi, we can add multiples of 2pi to the angle until we get a positive angle. In this case, we add 4pi (since 4pi = 12pi/3) to get the equivalent positive angle:
  (-10pi/3) + (12pi/3) = 2pi/3.

2. Find the cosine value of the positive angle: Now, we find the cosine value of the positive angle 2pi/3. Using the unit circle, we can determine that cos(2pi/3) = -1/2.

So, the exact value of the expression cos(-10pi/3) is -1/2.

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We can express 0.19 as the ratio of integers 1919/10000 and the repeating decimal 0.19191919... as the ratio of integers 1919/1000000.

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All the midpoints lie on a vertical line passing through x = 4. This means that the line of reflection is the vertical line x = 4, which corresponds to a reflection across the y-axis.

A polygon is a closed, two-dimensional structure made up of three or more straight line segments in geometry.

Each line segment forms an angle with the next one, and the point where two segments meet is called a vertex of the polygon.

The most common polygons are triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), heptagons (7 sides), octagons (8 sides), and so on. Polygons with more than 10 sides are usually named using the Greek numerical prefixes (e.g., a 12-sided polygon is called a dodecagon).

To determine the line of reflection that maps the polygon ABCDE onto the polygon A' B' C' D' E', we need to identify a line that is equidistant from the corresponding vertices of both polygons. We can start by finding the midpoint between each pair of corresponding vertices:

Midpoint between A(-3, 3) and A'(11, 3) is ((-3 + 11)/2, (3 + 3)/2) = (4, 3)

Midpoint between B(-3, 6) and B'(11, 6) is ((-3 + 11)/2, (6 + 6)/2) = (4, 6)

Midpoint between C(1, 6) and C'(7, 6) is ((1 + 7)/2, (6 + 6)/2) = (4, 6)

Midpoint between D(1, 3) and D'(7, 3) is ((1 + 7)/2, (3 + 3)/2) = (4, 3)

Midpoint between E(-1, 1) and E'(9, 1) is ((-1 + 9)/2, (1 + 1)/2) = (4, 1)

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The 99% confidence interval is:

p-hat ± z*SE = = (0.4716, 0.5664)

We can use the formula for calculating confidence intervals for a proportion:

p-hat ± z*SE

where p-hat is the sample proportion, SE is the standard error, and z is the z-score corresponding to the desired level of confidence.

For a 95% confidence interval, the z-score is 1.96 (from a standard normal distribution table).

Using the given values, we have:

p-hat ± z*SE = 0.519 ± 1.96(0.0184) = (0.4829, 0.5551)

To find the 99% confidence interval, we need to use a z-score of 2.576 (from the standard normal distribution table).

So, the 99% confidence interval is:

p-hat ± z*SE = 0.519 ± 2.576(0.0184) = (0.4716, 0.5664)

Therefore, the answer is B. (0.4716, 0.5664).

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The equation of the ellipse centered at (4, 1) with a minor axis of 8 units and a major axis of 10 units parallel to the x-axis is: (x - 4)²/25 + (y - 1)²/16 = 1

To write the equation of the ellipse centered at (4, 1) with a minor axis of 8 units, a major axis of 10 units, and parallel to the x-axis.
We will use the standard equation of an ellipse in the form:
(x - h)²/a² + (y - k)²/b² = 1
Here, (h, k) represents the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis.
Given that the ellipse is centered at (4, 1), we have h = 4 and k = 1.

Since the major axis is 10 units long and parallel to the x-axis, the semi-major axis a is half of that, which is 5 units.

Similarly, the minor axis is 8 units long, so the semi-minor axis b is half of that, which is 4 units.
Now, we can plug these values into the standard equation of an ellipse:
(x - 4)²/5² + (y - 1)²/4² = 1
Simplify the equation to:
(x - 4)²/25 + (y - 1)²/16 = 1
The equation of the ellipse centered at (4, 1) with a minor axis of 8 units and a major axis of 10 units parallel to the x-axis is:
(x - 4)²/25 + (y - 1)²/16 = 1

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The amount of fencing, dirt and sod for the baseball field are: length of Fencing & 1410.5 ft. Area of the sod ≈ 118017.13ft² Area of the field covered with distance ≈ 7049.6ft²

Area of a circle = πr²

Circumference of a circle = 2πr

where r is the radius of the circle

The area of a Quarter of a circle is therefore;

Area of a circle/ 4

The perimeter of a Quarter of a Circle is;

The perimeter of a circle/4

Fencing = ¼ x 2 x π x 380 + 2 x 15 +2 x 380 + ¼ x 2 x π x 15

Fencing = 197.5π + 190π = 1410.5 feet.

Grass =

π/4 x (380 - 6)² + 87 ² - π/4 × (87 + 30)² + 2 x 380 x 15 + π/4 x 15² - (3/4) x π x 10² - 25π

= 31528π + 18969 = 118017.13

The area Covered by the sod is about 118017.13Sq ft.

Dirt = π/4 x 380 ² - π/4 x (380 - 6)² + π/4 (87 + 30)² - 87² + π100 = (18613π - 30276)/4

= 7049.6

Therefore, the area occupied by the dirt is about 7049.6 Sq ft.

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The general solution to the homogeneous differential equation (d²y/dt²)−18(dy/dt)+97y=0 is y(t) = C₁ [tex]e^3^t[/tex] cos(8t) + C₂ [tex]e^3^t[/tex]sin(8t).

To solve the given differential equation, first, we need to find the characteristic equation by replacing d²y/dt² with r², dy/dt with r, and y with 1. This gives us the quadratic equation r² - 18r + 97 = 0.

Next, find the roots of the characteristic equation using the quadratic formula, which yields r = 3 ± 8i.

Since the roots are complex conjugates, the general solution to the homogeneous differential equation takes the form y(t) = [tex]e^\alpha^t[/tex](C₁cos(βt) + C₂sin(βt)), where α and β are the real and imaginary parts of the complex roots, respectively. In this case, α = 3 and β = 8. Substituting these values, we obtain the general solution y(t) = C₁ [tex]e^3^t[/tex]cos(8t) + C₂ [tex]e^3^t[/tex]sin(8t).

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The painting is 75 cm long, if the painting is 45 cm wide.

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

Ratio of the length to the width is 5:3. T

This means that

Length : Width = 5 : 3

The painting is 45 cm wide.

So, we have

Length : 45 = 5 : 3

Express as a fraction

So, we have

Length/45 = 5/3

Evaluate the above expression

so, we have the following representation

Length = 75

Hence, the length is 75

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Okay, here are the steps to solve this problem:

* There are 10 tenors, 9 sopranos, 20 altos, and 14 basses in the choir

* In total there are 10 + 9 + 20 + 14 = 53 singers

* There are 9 sopranos out of the 53 total singers

* To find the probability of a randomly selected singer being a soprano:

* Probability = (Number of desired outcomes) / (Total possible outcomes)

* So Probability = 9/53

Therefore, the probability that a randomly selected singer will be a soprano is 9/53

Answer:

  y(x) = -(2/e)x +3

Step-by-step explanation:

You want the equation of the line tangent to the parametric curve at t=1.

  (x, y) = (e^(√t), t -2·ln(t))

At t=1, the point of tangency is ...

  (x, y) = (e^(√1), 1 -2·ln(1)) = (e, 1)

The derivatives with respect to t are found using the chain rule:

  dx = d(e^u)du = d(e^√t)(1/(2√t))dt

  dx = (e^√t)/(2√t))·dt

  dy = (1 -2/t)·dt

Then the slope of the tangent line is ...

  m = dy/dx = (1 -2/t)(2√t)/e^√t

For t=1, this is ...

  m = (1 -2/1)(2√1)/(e^1) = -2/e

The equation for a line with slope m through point (h, k) is ...

  y = m(x -h) +k

The equation for a line with slope -2/e through point (e, 1) is ...

  y = (-2/e)(x -e) +1

  y = (-2/e)x +3

Answer:

  y(x) = -(2/e)x +3

Step-by-step explanation:

You want the equation of the line tangent to the parametric curve at t=1.

  (x, y) = (e^(√t), t -2·ln(t))

At t=1, the point of tangency is ...

  (x, y) = (e^(√1), 1 -2·ln(1)) = (e, 1)

The derivatives with respect to t are found using the chain rule:

  dx = d(e^u)du = d(e^√t)(1/(2√t))dt

  dx = (e^√t)/(2√t))·dt

  dy = (1 -2/t)·dt

Then the slope of the tangent line is ...

  m = dy/dx = (1 -2/t)(2√t)/e^√t

For t=1, this is ...

  m = (1 -2/1)(2√1)/(e^1) = -2/e

The equation for a line with slope m through point (h, k) is ...

  y = m(x -h) +k

The equation for a line with slope -2/e through point (e, 1) is ...

  y = (-2/e)(x -e) +1

  y = (-2/e)x +3

An implicit equation for the plane passing through the points (-5,0,5), (-5,-5,0), and (-8,-5,10) is x - 5y + 3z - 5 = 0.

To find the equation of a plane passing through three points, we can use the following formula:

(x - x1)(y2 - y1)(z3 - z1) + (y - y1)(z2 - z1)(x3 - x1) + (z - z1)(x2 - x1)(y3 - y1) = (x2 - x1)(y3 - y1)(z3 - z1) + (y2 - y1)(z3 - z1)(x3 - x1) + (z2 - z1)(x3 - x1)(y3 - y1)

where (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) are the given points.

Substituting the given values, we get:

(x + 5)(-5)(10) + (y - 0)(-5)(-8) + (z - 5)(-5)(0) = (y + 5)(-5)(10) + (z - 0)(-5)(-8) + (x + 5)(-5)(0)

Simplifying this equation, we get:

-50x + 50y - 50z + 250 = 0

Dividing both sides by -50, we get:

x - 5y + 3z - 5 = 0

Hence, the implicit equation for the plane passing through the points (-5,0,5), (-5,-5,0), and (-8,-5,10) is x - 5y + 3z - 5 = 0.

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