PLS HELP ASAP!
Given F(x) = x- 4

What is the zero of this function?

-6
-3/2
6

Models should be produced of the function C(x) = 10 + 2 x is 329.4 .

Cost of manufacturing is

C(x) = 10 + 2 x

Sale price in soles of each model is

P(x) = 20 - [tex]\frac{6x^{2} }{800}[/tex]

U(x) is the utility function

U(x) = x P(x) - C(x)

U(x) = x (20 - [tex]\frac{6x^{2} }{800}[/tex]  ) - (10 +2x)

U(x) = 20x - [tex]\frac{6x^{3} }{800}[/tex]   - 10 - 2x

U(x) = 18x - [tex]\frac{6x^{3} }{800}[/tex] - 10

U'(x) = 18 - 18x²/800

For maximum model U'(x) = 0

18 - 18x²/800 = 0

18x²/800 = 18

x² = 800

x = √800

x = 20√2

U(x) = 18(20√2 ) - [tex]\frac{6(20\sqrt{2} )^{2} }{800}[/tex] - 10

U(x) = 329.5

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The question is in Spanish question in English :

The manufacturing costs of models are modeled to the following function. C(x) = 10 + 2x. The manufacturer estimates that the sale price in soles of each model is given by: P(x) = 20- 6x2/800 How many models should be produced?

There are 8 major routes from the center of Happy Town to the center of Peaceful Town that go through the center of Miserable Town, we need to use the concept of permutations and combinations.

There are 5 major routes from Happy Town to Miserable Town, and 3 major routes from Miserable Town to Peaceful Town. Therefore, there are a total of 5 x 3 = 15 possible routes from Happy Town to Peaceful Town via Miserable Town. However, not all of these routes are unique. Some of them may overlap or follow the same path. To eliminate these duplicates, we need to consider the routes that start from Happy Town, pass through Miserable Town, and end at Peaceful Town as a group. Since there are 5 routes from Happy Town to Miserable Town, we can choose any one of them as the starting point. Similarly, since there are 3 routes from Miserable Town to Peaceful Town, we can choose any one of them as the ending point. Therefore, there are 5 x 3 = 15 possible combinations of starting and ending points. However, we have counted each route twice, once for each direction. So, we need to divide the total number of combinations by 2 to get the final answer. Therefore, the number of major routes from the center of Happy Town to the center of Peaceful Town that go through the center of Miserable Town is 15 / 2 = 7.5. However, since we cannot have half a route, we round up to the nearest whole number.

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The height of the pole would be 29.94 feet.

The scenario results in the formation of a right angle triangle.

The length of the support wire represents the hypotenuse of the right angle triangle.

The ground distance between the wire and the pole represents the adjacent side of the right angle triangle.

The height of the pole represents the opposite side of the right angle triangle.

To determine the height of the pole, h, we would apply Pythagoras theorem

Hypotenuse² = opposite side² + adjacent side²

18² = 10² + h²

324 = 100 + h²

h² = 324 - 100 = 224

h = √224 = 14.97

The wire meets the pole halfway up the pole.

The height of the pole would be

14.97 × 2 = 29.94 feet

Hence, The height of the pole would be 29.94 feet.

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complete question:

An 18 foot support wire is attached to a vertical pole. The wire is attached to the ground 10 feet away from the pole. If the wire meets the pole halfway up the pole, how y'all is the pole in feet?

Answer:

18ft

Step-by-step explanation:

6/27= 4.5, 27/4. 4.5x4=18

Answer:

18ft

Step-by-step explanation:

I am taking the test right now and I think this would be the correct answer!

A simple way I found out: 6/4 = 1.5 so I took 1.5 and divided 27 by it.  27/1.5 = 18

Hope this helped!

The given joint probability mass function defines the probabilities of the discrete random variables x and y taking on values 1, 2, or 3.

The probability p(x = x, y = y) is equal to xy/54 for all (x, y) in the set {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.  To find the marginal probability mass functions for x and y, we sum the joint probabilities over all possible values of the other variable. That is,

p(x = x) = ∑ p(x = x, y = y) = ∑ xy/54, y=1 to 3

         = (x/54)∑y=1 to 3 y

         = (x/54)(1+2+3)

         = (x/54)(6)

         = x/9

Similarly, we have

p(y = y) = ∑ p(x = x, y = y) = ∑ xy/54, x=1 to 3

         = (y/54)∑x=1 to 3 x

         = (y/54)(1+2+3)

         = (y/54)(6)

         = y/9

Hence, the marginal probability mass functions for x and y are given by p(x) = x/9 and p(y) = y/9, respectively.

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A point on the edge of a 45 rpm record is moving 0.18 ft/sec faster than a point on the edge of a 33 rpm record.

To compare the speeds of the two records, we need to find the linear velocity of a point on the edge of each record. The linear velocity is the distance traveled by a point on the edge of the record in a given time.

For the 45 rpm record, the diameter is 7 inches, which means the radius is 3.5 inches (7/2). The circumference of the record is then 2πr = 2π(3.5) = 22 inches. To convert this to feet, we divide by 12 to get 1.83 feet.

The linear velocity of a point on the edge of the 45 rpm record is then:

V45 = 1.83 ft/circumference x 45 rev/min x 1 min/60 sec = 1.91 ft/sec

For the 33 rpm record, the diameter is 12 inches, which means the radius is 6 inches (12/2). The circumference of the record is then 2πr = 2π(6) = 37.7 inches. To convert this to feet, we divide by 12 to get 3.14 feet.

The linear velocity of a point on the edge of the 33 rpm record is then:

V33 = 3.14 ft/circumference x 33 rev/min x 1 min/60 sec = 1.73 ft/sec

The difference in speeds between the two records is then:

V45 - V33 = 1.91 ft/sec - 1.73 ft/sec = 0.18 ft/sec

Therefore, a point on the edge of a 45 rpm record is moving 0.18 ft/sec faster than a point on the edge of a 33 rpm record.

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Bring to slope intercept form

As per y=MX+c form

m is slope

Now perpendicular lines have slope negative reciprocal to each other

so slope of perpendicular line

Now equation of line

Suppose that f(x) = −3x4 is an antiderivative of f(x) and g(x) = 2x3 is an antiderivative of g(x). The valsue of  ∫(f(x) g(x))dx is " -108/5 + C", where C is the constant of integration.

To find the integral of the product of two functions, we can use the formula ∫(f(x) g(x))dx = f(x) ∫g(x)dx - ∫[f'(x) (∫g(x)dx)]dx. Using this formula, we can evaluate the given integral as follows:

∫(f(x) g(x))dx = (-3x^4)(2x^3) - ∫[(-12x^3)(2x^3)]dx = -6x^7 + 24x^6/2 + C = -108/5 + C.

Therefore, the answer is " -108/5 + C".

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= 0 - Var(y) = 0  Thus, we have shown that x - y and xy are uncorrelated.

To show that x - y and xy are uncorrelated, we need to show that their covariance is zero.

Covariance is defined as Cov(x,y) = E[(x - E[x])(y - E[y])].

We can expand the covariance of x-y and xy as follows:

Cov(x - y, xy) = E[(x - y - E[x - y])(xy - E[xy])]

= E[((x - E[x]) - (y - E[y]))(xy - E[x]E[y])]

= E[(x - E[x])xy - (x - E[x])E[y] - (y - E[y])E[x] + (y - E[y])E[x]E[y]]

= E[(x - E[x])xy] - E[(x - E[x])E[y]] - E[(y - E[y])E[x]] + E[(y - E[y])E[x]E[y]]

= E[(x - E[x])xy] - E[x - E[x]]E[y - E[y]] - E[y - E[y]]E[x - E[x]] + E[x - E[x]]E[y - E[y]]

= E[(x - E[x])xy] - (Var(x) - 0)

= E[(x - E[x])xy] - Var(x)

Since x and y have the same variance, Var(x) = Var(y). Therefore:

Cov(x - y, xy) = E[(x - E[x])xy] - Var(x) = E[(x - E[x])xy] - Var(y)

To show that this is equal to zero, we can use the fact that E[xy] = E[x]E[y] since x and y are independent.

Therefore:

Cov(x - y, xy) = E[(x - E[x])xy] - Var(y)

= E[(x - E[x])y]E[x] - E[(x - E[x])E[y]] - Var(y)

= E[(x - E[x])](E[xy] - E[x]E[y]) - Var(y)

= 0 - Var(y) = 0

Thus, we have shown that x - y and xy are uncorrelated.

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The fraction of tomatoe recipe greater than 1 used is 7/4

The question requires that we Subtract the required value of tomato recipe from 1

The subtraction expression can be written thus ;

Amount of tomatoe recipe - 1

[tex]2 \frac{3}{4} - 1[/tex]

Converting to a proper fraction;

11/4 - 1/1

Take L.C.M of the denominator

L.C.M of 4 and 1 is 4

11/4 - 1/1 = (11 - 4)/4

11/4 - 1/1 = 7/4

Therefore, the fraction of tomatoe greater than 1 used is 7/4

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The gradient of the function at the point (3, 8) is given by the vector (-5/7, -3/49), and the maximum rate of change of the function at this point is sqrt(354/2401).

The gradient of a function is a vector that points in the direction of the maximum rate of change of the function and its magnitude gives the rate of change at that point. To find the gradient of the function z = ln(x^2 - y)x - 1 at the point (3, 8), we need to take the partial derivatives of z with respect to x and y, and evaluate them at the point (3, 8).

The partial derivative of z with respect to x is given by (2x - y)/(x^2 - y) and the partial derivative of z with respect to y is -x/(x^2 - y). Therefore, the gradient of z is given by the vector:

∇z = [(2x - y)/(x^2 - y)] i - [x/(x^2 - y)] j

We can now evaluate this gradient vector at the point (3, 8) by substituting x = 3 and y = 8:

∇z(3, 8) = [-5/7] i - [3/49] j

This tells us that the maximum rate of change of the function at the point (3, 8) is in the direction of the vector [-5/7, -3/49], and the rate of change in this direction is given by the magnitude of the gradient vector, which is |∇z(3, 8)| = sqrt((25/49) + (9/2401)) = sqrt(354/2401).

So the gradient of the function at the point (3, 8) is given by the vector (-5/7, -3/49), and the maximum rate of change of the function at this point is sqrt(354/2401).

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There are different questions that can be asked regarding this scenario, but one common question is: what is the probability that all four chocolates have coconut filling?

To answer this question, we can use the hypergeometric distribution, which describes the probability of obtaining a certain number of "successes" (in this case, chocolates with coconut filling) in a sample of a given size (in this case, four) taken from a population of a given size (in this case, 16 chocolates with four of them having coconut filling), without replacement. The probability of getting four chocolates with coconut filling is then:

P(X = 4) = (4 choose 4) * (16 - 4 choose 0) / (16 choose 4) = 1/182

where "n choose k" denotes the number of ways of choosing k items from a set of n items, and the probability of each possible sample is the same. Therefore, the probability of all four chocolates having coconut filling is very low, only about 0.55%.

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Answer:

4/3

Step-by-step explanation:

Divide 1 by 3/4 to find out how many groups.

1 divided by 3/4 is 4/3

To the nearest foot, the person would fall approximately 20 feet before hitting the ground. This highlights the importance of using ladders safely and following proper safety protocols.

Ladders are indeed dangerous if not used correctly, and accidents can happen even with the slightest miscalculation or carelessness. In this scenario, we have a 20 ft extension ladder placed on a wall, making an angle of elevation of 85 degrees with the ground. If the person at the top of the ladder leaned back, rotating the ladder away from the wall, we need to determine how far they would fall before hitting the ground.
To solve this problem, we need to use trigonometry. The angle of elevation is 85 degrees, which means the complementary angle is 5 degrees. We can use the tangent function to find the length of the ladder that is off the wall, which is the height the person will fall from.
tan(5) = height of ladder off the wall / length of ladder
Length of ladder = 20 ft
Height of ladder off the wall = tan(5) x 20 = 1.75 ft
Therefore, if the person at the top of the ladder leaned back and rotated it away from the wall, they would fall approximately 1.75 ft before hitting the ground. It is important to always follow ladder safety guidelines and use caution when using a ladder to avoid accidents and injuries.
Using a ladder can indeed be dangerous if not used correctly. In this scenario, we have a 20 ft extension ladder placed on a wall at an angle of elevation of 85 degrees. To determine how far the person would fall before hitting the ground when the ladder rotates away from the wall, we'll need to use trigonometry.
Step 1: Identify the known values.
- The ladder length (hypotenuse) is 20 ft.
- The angle of elevation is 85 degrees.
Step 2: Determine the height of the ladder when it's placed against the wall.
- We can use the sine function to find the height: sin(angle) = height / ladder_length.
- Plug in the known values: sin(85) = height / 20.
Step 3: Solve for the height.
- Multiply both sides by 20: height = 20 * sin(85).
- Calculate the height: height ≈ 19.98 ft.
Step 4: Determine the distance the person falls.
- The person falls from the height of the ladder to the ground, so the falling distance is approximately 19.98 ft.
To the nearest foot, the person would fall approximately 20 feet before hitting the ground. This highlights the importance of using ladders safely and following proper safety protocols.

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The required solution to the system of equations in Question 11 is x = -6 and y = -4, and in Question 12 is x = -7 and y = 9.

11:

3x + 2y = -26

4x - 5y = -4

To eliminate one variable, we can multiply the first equation by 4 and the second equation by 3, so the coefficients of x will be the same:

12x + 8y = -104 (Multiplying the first equation by 4)

12x - 15y = -12 (Multiplying the second equation by 3)

Now, subtract the second equation from the first equation to eliminate x:

(12x + 8y) - (12x - 15y) = -104 - (-12)

12x + 8y - 12x + 15y = -104 + 12

y = -4

Now, substitute the value of y back into one of the original equations, let's use the first equation:

3x + 2(-4) = -26

3x = -18

x = -6

Therefore, the solution to the system of equations in Question 11 is x = -6 and y = -4.

Similarly,

The solution to the system of equations in Question 12 is x = -7 and y = 9.

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The outcome of the survey would be biased if the plan for selecting the 50 members to participate in the survey is not representative of the entire membership of 250 people. Several scenarios could introduce bias into the survey:

Convenience Sampling: If the surveyors simply approach the first 50 members they encounter at the club, it would introduce bias because it assumes all members have an equal chance of being selected. However, this method may inadvertently exclude certain groups, such as those who frequently play during specific time slots.

Self-Selection Bias: If the survey is conducted on a voluntary basis, where members can choose whether to participate, it can introduce self-selection bias. Members who have extreme opinions, either highly satisfied or dissatisfied, may be more likely to participate, leading to an inaccurate representation of the overall satisfaction levels.

Demographic Bias: If the surveyors do not consider the demographic diversity within the club while selecting participants, it may result in biased outcomes. For example, if the survey predominantly includes only male or only female members, it may not accurately represent the satisfaction levels of both genders.

To avoid bias, it is crucial to use a random sampling method that ensures each member has an equal chance of being selected for the survey. This way, the selected sample will more accurately reflect the overall satisfaction of the entire membership.

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The annual interest rate on the $5600 loan, with $1008 of interest accrued over 4 years, is 4.5%, as calculated using the formula for simple interest. Option B.

To find the annual interest rate, we can use the formula for simple interest: I = P * R * T, where I is the interest, P is the principal amount (loan amount), R is the interest rate, and T is the time in years.

Given that the loan amount is $5600 and the interest after 4 years is $1008, we can rearrange the formula to solve for R. In this case, R = (I / P) / T = (1008 / 5600) / 4 = 0.045 = 4.5%. Therefore, the annual interest rate on the loan is 4.5%. The correct answer is option (b) 4.5%.

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Answer:

The answer is 4

Step-by-step explanation:

Marked price: 39 dollars

Selling price: 31 dollars

Discount: ???

Subtract the post-discount price from the pre-discount price.

31-39=-8

Divide this new number by the pre-discount price.

-8/39=-0.20512820512

Multiply the resultant number by 100.

-0.20512820512×100= -20.512820512

Round the number

-20.512820512 → -20.5

The discount was %20.5 off

I hoped I solved your question, if I did not you can tell me and I would be more than glad to fix it ૮ ˶ᵔ ᵕ ᵔ˶ ა

Answer:

$1,134 + $926 + $562 + $333 + $55 + $130

+ $1,500 + $313 = $4,953

The length of the curve is:

[tex]=\frac{(68)^\frac{3}{2}-8 }{3}[/tex]

Using the arc length formula in terms of polar coordinates [tex]\int\limits\sqrt{r^2+(\frac{dr}{d\theta})^2 }[/tex]

To find the length of the curve we will use the formula:

[tex]\int\limits\sqrt{r^2+(\frac{dr}{d\theta})^2 }[/tex]

Now, Let us put it in the expression:

[tex]r = \theta^2\\\\\frac{dr}{d\thera} =2\theta[/tex]

Now the integral becomes:

[tex]=\int\limits^8_0 \sqrt{(\theta)^4+(2\theta)^2} \, d\theta\\ \\=\int\limits^8_0\theta \sqrt{(\theta)^2+4} \, d\theta\\[/tex]

Now using the substitution method:

[tex]\theta^2+4=t\\\\2\thetad\theta=dt\\\\=\int\limits\frac{\sqrt{t}dt }{2}\\ \\=\frac{t^\frac{3}{2} }{3}\\ \\=\frac{(\theta^2+4)^\frac{3}{2} }{3}[/tex]

Now, Let us plug in the values:

[tex]=\frac{(68)^\frac{3}{2}-8 }{3}[/tex]

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If u is an orthogonal matrix and v is constructed by interchanging some of the columns of u, then v is also an orthogonal matrix. This is because the columns of an orthogonal matrix are orthonormal.

An orthogonal matrix is a square matrix whose columns are orthonormal. This means that each column has a length of 1 and is orthogonal to all the other columns. Formally, this can be written as:

u^T u = u u^T = I

where u^T is the transpose of u and I is the identity matrix.

Now suppose we construct a new matrix v by interchanging some of the columns of u. Let's say we interchange columns j and k, where j and k are distinct column indices of u. Then the matrix v is given by:

v = [u_1, u_2, ..., u_{j-1}, u_k, u_{j+1}, ..., u_{k-1}, u_j, u_{k+1}, ..., u_n]

where u_i is the ith column of u.

To show that v is orthogonal, we need to show that its columns are orthonormal. Let's consider the jth and kth columns of v. By construction, these columns are u_k and u_j, respectively, and we know from the properties of u that:

u_j^T u_k = 0 and u_j^T u_j = u_k^T u_k = 1

Therefore, the jth and kth columns of v are orthogonal and have a length of 1, which means they are orthonormal. Moreover, all the other columns of v are also orthonormal because they are simply copies of the corresponding columns of u, which are already orthonormal.

Finally, we can show that v is indeed an orthogonal matrix by verifying that v^T v = v v^T = I, using the definition of v and the properties of u. This completes the proof that v is an orthogonal matrix.

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a. To select a committee of three men and four women, we can choose three men from six distinct men and four women from seven distinct women. This can be done in:

C(6, 3) * C(7, 4) = 20 * 35 = 700 ways.

Therefore, there are 700 ways to select a committee of three men and four women.

b. To select a committee of four persons that has at least one woman, we can either choose one woman and three men or choose two women and two men or choose three women and one man or choose four women. We can calculate the number of ways for each case and add them up to get the total number of ways.

One woman and three men: C(7, 1) * C(6, 3) = 7 * 20 = 140 ways

Two women and two men: C(7, 2) * C(6, 2) = 21 * 15 = 315 ways

Three women and one man: C(7, 3) * C(6, 1) = 35 * 6 = 210 ways

Four women: C(7, 4) = 35 ways

The total number of ways to select a committee of four persons that has at least one woman is the sum of the above cases:

140 + 315 + 210 + 35 = 700 ways.

Therefore, there are 700 ways to select a committee of four persons that has at least one woman.

c. To select a committee of four persons that has persons of both sexes, we can choose two men from six distinct men and two women from seven distinct women or choose three men from six distinct men and one woman from seven distinct women or choose one man from six distinct men and three women from seven distinct women. We can calculate the number of ways for each case and add them up to get the total number of ways.

Two men and two women: C(6, 2) * C(7, 2) = 15 * 21 = 315 ways

Three men and one woman: C(6, 3) * C(7, 1) = 20 * 7 = 140 ways

One man and three women: C(6, 1) * C(7, 3) = 6 * 35 = 210 ways

The total number of ways to select a committee of four persons that has persons of both sexes is the sum of the above cases:

315 + 140 + 210 = 665 ways.

Therefore, there are 665 ways to select a committee of four persons that has persons of both sexes.

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There are 360 ways for a four-letter words that can be formed using the letters of the word finite. So, correct option is B.

To find the number of four-letter words that can be formed using the letters of the word "finite," we can use the permutation formula, which is:

nPr = n! / (n-r)!

where n is the total number of items to choose from, and r is the number of items to choose. In this case, we have 6 letters to choose from (n=6), and we want to choose 4 letters (r=4).

Therefore, the number of four-letter words that can be formed is:

6P₄ = 6! / (6-4)!

= 6! / 2!

= (6 x 5 x 4 x 3 x 2 x 1) / (2 x 1)

= 720 / 2

= 360

Therefore, the answer is 360, which corresponds to option B.

In summary, there are 360 four-letter words that can be formed using the letters of the word "finite," by using the permutation formula to calculate the number of possible arrangements of the 6 letters taken 4 at a time.

So, correct option is B.

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Answer:

x = 46

Step-by-step explanation:

These 2 angles would add up to 180 degrees.

So 41 + (3x+1) = 180

Let's solve for x.

41 + 3x + 1 = 180

Combine like terms.

42 + 3x = 180

subtract 42 from both sides

3x = 180-42

3x = 138

divide both sides by 3

x = 46

Answer:

We can solve the simultaneous equation 24n + 9m = 8 and 3n - 2m = 6 by using the elimination method.

First, we need to multiply the second equation by 3 to eliminate n:

24n + 9m = 8

(3n - 2m) × 3 = 6 × 3

9n - 6m = 18

Now we have two equations with the same n coefficient, so we can subtract the second equation from the first to eliminate n:

24n + 9m = 8

-(9n - 6m = 18)

-----------------

15n + 15m = -10

We can simplify this equation by dividing both sides by 5:

3n + 3m = -2

Now we have two equations with the same m coefficient, so we can subtract the second equation from the first to eliminate m:

24n + 9m = 8

-(3n + 3m = -2)

----------------

21n + 6m = 10

We can simplify this equation by dividing both sides by 3:

7n + 2m = 10/3

Now we have two equations with only one variable, so we can solve for one variable and substitute the value into one of the original equations to solve for the other variable:

7n + 2m = 10/3

2m = 10/3 - 7n

m = (10/3 - 7n)/2

Substitute this expression for m into the first equation:

24n + 9m = 8

24n + 9[(10/3 - 7n)/2] = 8

24n + (30/2 - 63n/2)/2 = 8

24n + 15 - 63n/4 = 8

24n - 63n/4 = 8 - 15

(96n - 63n)/4 = -7

33n/4 = -7

n = -28/33

Substitute this value of n into the second equation:

3n - 2m = 6

3(-28/33) - 2m = 6

-28/11 + 2m/11 = 2

2m/11 = 2 + 28/11

2m/11 = 50/11

Answer:

n = 14 / 15
m = -8 / 5

Step-by-step explanation:

24n + 9m = 8   ------- (1)   x 2

3n - 2m = 6  -----------(2)   x 9

48n + 18m = 16    ------- (3)
27n - 18m = 54    --------(4)

Adding two eqn , we get ;
______________

75n = 70
n = 14 / 15

Putting value of n in eqn (2) , we get ;

14 / 5 - 2m = 6
2m = 14 / 5 - 6

2m = -16 / 5

m = -8 / 5

The measure of the various angles are:

∠1= 130°

∠3 = 130°

∠5= 50°

∠6= 130°

Lines m and l are the parallel lines and a line 'n' is a transverse intersecting these lines.

m∠2 = 50°

m∠1 + m∠2 = 180° [Linear pair of angles]

m∠1 = 180° - 50°

m∠1 = 130°

m∠3 = m∠1 = 130° [Vertically opposite angles]

m∠3 + m∠5 = 180° [Consecutive interior angles]

m∠5 = 180° - m∠3

       = 180° - 130°

m∠5   = 50°

m∠6 + m∠5 = 180° [Linear pair of angles]

m∠6 = 180° - 50°

m∠6= 130°

Hence,  the measure of the various angles are:

∠1= 130°

∠3 = 130°

∠5= 50°

∠6= 130°

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Full Question:

Although part of your question is missing, you might be referring to this full question:

See the attached image.

C divides AB into two congruent segments, AC and CB, we can say that C is the midpoint of AB.

To prove that C is the midpoint of AB, we need to show that AC = CB and that C divides AB into two congruent segments.
We are given that AB = 12 and AC = 6.

Using the segment addition property, we can add AC and CB to get AB:

AC + CB = AB.
We can substitute the values we were given: 6 + CB = 12.

To find CB, we can subtract 6 from both sides of the equation: CB = 6.
Now we know that AC = 6 and CB = 6.

By the symmetric property, we can see that AC = CB.
Since AC and CB are congruent, we can use the definition of congruent segments to show that AC ≅ CB.
Finally, we can conclude that C is the midpoint of AB because it divides AB into two congruent segments, AC and CB. Therefore, we have proven that C is the midpoint of AB.
In summary, we used the segment addition property, substitution property, symmetric property, and definition of congruent segments to show that C is the midpoint of AB.
Using the segment addition property, we have AC + CB = AB.

Substituting the given values, we get 6 + CB = 12.

Using the subtraction property, we find CB = 6.
Now, we have AC = 6 and CB = 6. Since AC and CB have equal lengths, we can conclude that AC ≅ CB by the definition of congruent segments.
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