Graph the system below and write its solution
Y=1/4x-3
-x+4y=-4

The value of the constant c that makes the following function are c = 0.

Constant is a term used to describe a value that remains unchanged or fixed throughout a program or process. It can be a numeric value, a character value, a string, or a Boolean (true/false) value. Common examples of constants include physical constants, mathematical constants, and programming-language keywords.A constant is a value that does not change, regardless of the conditions or context in which it is used. Common examples of constants include mathematical values such as pi (3.14159), physical constants such as the speed of light (299,792,458 m/s), and other constants such as the universal gravitational constant (6.67408 × 10−11 m3 kg−1 s−2).

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Therefore, c must equal 0 in order for the two sides of the function to be equal. and The one with the greater absolute value is b = 10.

A function is a block of code that performs a specific task. It is a subprogram or a set of instructions that can be used multiple times in a program.

27. For the function to be continuous at x = 7, the limit of the function as x approaches 7 from the left must equal the limit of the function as x approaches 7 from the right.
This means that the value of y as x approaches 7 must be the same on both the left and right sides of the point.
Since the left side of the function is y = c*y + 3, the right side of the function must also be equal to y = c*y + 3.
Therefore, c must equal 0 in order for the two sides of the function to be equal.

28. In order for the function to be continuous at x = 5, the value of y at x = 5 must be the same on both the left and right sides of the point.
Since the left side of the function is y = b - 2x, the right side of the function must also be equal to y = b - 2x.
Therefore, b must equal 10 in order for the two sides of the function to be equal.

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Complete Question:

The length of the third side of a triangle provided if the first two sides measure 10 and 12 is 2 < c < 22.

Three sides and three angles make up each triangle. These triangle sides are straight line segments that meet at each of the triangle's vertices to produce a closed three-sided shape. Each side of a right-angled triangle is given a name. The hypotenuse of a right-angled triangle is its longest side, the base is its lowest side, and the perpendicular, which stands next to the right angle, is its standing line.

Each valid triangle has a side length that is smaller than the sum of the other two.

For a triangle whose sides are provided as positive integers a, b, and c, the following three inequalities result:

a+b > c, b+c > a, c+a > b.

It is also readable as

|a-b| < c < a+b.

Note that these inequalities can become equalities if you take into account degenerate triangles, where all of the vertices are on the same line (collinear), but since degenerate triangles aren't actually triangles, I didn't include them.

In response to your query, we may calculate c as follows if a = 10 and

b = 12:

2 < c < 22,

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Answer:

Step-by-step explanation:

To solve this problem, we need to use the equations of motion for constant acceleration, constant velocity, and constant deceleration. We'll break the problem into several parts and use these equations to find the distance traveled in each part. Then, we'll add up the distances to get the total distance traveled.

First, we need to convert the units of speed from km/h to m/s, since the equations of motion use meters per second. We have:

Initial speed (u) = 0 km/h = 0 m/s

Final speed (v) = 40 km/h = 11.11 m/s

Constant speed = 40 km/h = 11.11 m/s (for 4 minutes)

Final speed before hill = 30 km/h = 8.33 m/s

Speed at top of hill = 0 m/s

Acceleration (a) = (v-u)/t = (11.11-0)/(3*60) = 0.0611 m/s^2

PART 1: ACCELERATION PHASE

Time taken (t) = 3 minutes = 180 seconds

Distance traveled (s) = ut + (1/2)at^2

s = 0 + (1/2)0.0611(180^2) = 331.83 meters

PART 2: CONSTANT SPEED PHASE

Time taken (t) = 4 minutes = 240 seconds

Distance traveled (s) = vt

s = 11.11*240 = 2666.4 meters

PART 3: DECELERATION PHASE

Time taken (t) = 1 minute = 60 seconds

Deceleration (a) = (v-u)/t = (8.33-11.11)/60 = -0.0461 m/s^2 (negative since it's deceleration)

Distance traveled (s) = vt + (1/2)at^2

s = 8.3360 + (1/2)(-0.0461)*(60^2) = 494.7 meters

PART 4: CONSTANT SPEED PHASE

Time taken (t) = 10 minutes = 600 seconds

Distance traveled (s) = vt

s = 8.33*600 = 4998 meters

PART 5: DECELERATION PHASE TO STOP

Time taken (t) = 2 minutes = 120 seconds

Initial speed (u) = 8.33 m/s

Final speed (v) = 0 m/s

Deceleration (a) = (v-u)/t = (0-8.33)/120 = -0.0694 m/s^2

Distance traveled (s) = vt + (1/2)at^2

s = 8.33120 + (1/2)(-0.0694)*(120^2) = 733.3 meters

TOTAL DISTANCE TRAVELED:

Adding up the distances from each part, we get:

Total distance = 331.83 + 2666.4 + 494.7 + 4998 + 733.3 = 9184.23 meters = 9.18 km (rounded to two decimal places)

Therefore, the cyclist has traveled approximately 9.18 km.

Answer:

Step-by-step explanation:

To solve the initial-value problem y''' − 2y'' + y' = 2 − 24ex + 40e5x, y(0) = 1/2, y'(0) = 5/2, y''(0) = −11/2, we first find the characteristic equation by assuming that y = e^(rt):

r^3 - 2r^2 + r = 0

r(r^2 - 2r + 1) = 0

r(r-1)^2 = 0

r = 0, 1, 1

Therefore, the general solution to the homogeneous equation y''' − 2y'' + y' = 0 is:

y_h = c1 + c2e^x + c3xe^x

To find the particular solution to the non-homogeneous equation y''' − 2y'' + y' = 2 − 24ex + 40e5x, we guess a particular solution of the form:

y_p = Ax^2e^5x + Be^x

y_p' = (2Ax + 5Ax^2)e^5x + Be^x

y_p'' = (10Ax^2 + 4Ax + 25Ax^2)e^5x + Be^x

y_p''' = (70Ax^2 + 60Ax + 10A + 25Ax^2)e^5x + Be^x

Substituting these expressions into the original equation, we get:

(70Ax^2 + 60Ax + 10A + 25Ax^2)e^5x + Be^x − 2[(10Ax^2 + 4Ax + 25Ax^2)e^5x + Be^x] + [(2Ax + 5Ax^2)e^5x + Be^x] = 2 − 24ex + 40e5x

Simplifying, we get:

(45Ax^2 + 2Ax − 2)e^5x + Be^x = 2 − 24ex + 40e5x

Equating the coefficients of the like terms on both sides, we get:

45A = 40

2A − 2 = 0

B = 2

Therefore, the particular solution is:

y_p = 8/9 x^2e^5x + 2e^x

The general solution to the non-homogeneous equation is therefore:

y = c1 + c2e^x + c3xe^x + 8/9 x^2e^5x + 2e^x

Using the initial conditions y(0) = 1/2, y'(0) = 5/2, y''(0) = −11/2, we get:

c1 + c2 + 2 = 1/2

c2 + 2c3 + 5/2 = 5/2

2c2 + 10/9 + 10 = -11/2

Solving this system of equations, we get:

c1 = 1/9

c2 = -25/18

c3 = 0

Therefore, the solution to the initial-value problem y''' − 2y'' + y' = 2 − 24ex + 40e5x, y(0) = 1/2, y'(0) = 5/2, y''(0) = −11/2 is:

y = 1/9 - 25/18e^x + 8/9

Answer:

y > 0

Step-by-step explanation:

[tex]y = e^{4x}[/tex]

This is an exponential function;

Consider as x → ∞, y → ∞;

If x = 0. y = e⁰ = 1;

As x → -∞, y → 0;

y ranges from 0 to ∞ therefore, i.e. y > 0

The solution for b is: b = r * (f - h²).

the definition of an equation is a mathematical statement that demonstrates that two mathematical expressions are equal. For instance, the equation 3x + 5 = 14 consists of the two expressions 3x + 5 and 14, which are separated by the 'equal' symbol.

To solve for b, we need to isolate the variable b on one side of the equation. We can do that by multiplying both sides of the equation by (f - h²):

[tex]$\frac{b}{(f - h^2)} = r * (f - h^2)[/tex]

Now, we can isolate b by multiplying both sides by (f - h^2) again:

b = r * (f - h²)

Therefore, the solution for b is: b = r * (f - h²).

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The compound interest on ₹20,000 for 3 years at 10% per annum compounded annually is ₹6,620.

To calculate the compound interest on a principal amount of ₹20,000 for 3 years at 10% per annum compounded annually, we can use the formula

A = P(1 + R/100)^n

Where,

A = final amount after n years

P = principal amount

R = annual interest rate

n = number of years

In this case, P = ₹20,000, R = 10%, and n = 3 years.

So, applying the formula

A = 20000(1 + 10/100)^3

= 20000(1.1)^3

= 20000(1.331)

= ₹26,620

The final amount after 3 years is ₹26,620. Therefore, the compound interest is

Compound Interest = Final Amount - Principal Amount

= ₹26,620 - ₹20,000

= ₹6,620

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I have solved the question in general, as the given question is incomplete.

The complete question is:

Find the compound interest on ₹ 20,000 for 3 years at 10% per annum compounded annually.

In the given scenarios, Paired differences are option A, B, and D, the scenario which shows Differences between 2 independent samples is option C.

Paired differences refer to cases where the difference is measured within the same group of subjects or individuals, such as the difference in time spent with mother and father in a hetero family, or the difference in IQ between first-born and second-born twins.

Differences between 2 independent samples, on the other hand, refer to cases where the difference is measured between two separate groups, such as the difference in the number of books read per year by residents of California and New York.

Recognizing the type of difference is important in determining the appropriate statistical analysis to use in analyzing the data.

The answer option for paired are  A, B, and D and for Differences between 2 independent samples is option C.

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_____The given question is incomplete, the complete question is given below:

Which of the following cases represent paired differences and which are differences between 2 independent samples?

A The difference between the average time a child spends with their mother and the average time the child spends with their father (in a hetero family) [Select]

B The difference between the average IQ of the first-born twin and of the second-born twin [ Select]

C The difference between the average number of books read per year by the residents of the states of California and New York [Select] The difference between the average amount grossed by Paramount Pictures and Warner Bros. movies [Select ]

D The difference between the shares of the republican and the democratic candidate in each year of the US elections. [ Select ]

Answer:

There were 8 bears

Step-by-step explanation:

Letting L = number of lions, T = number of tigers and B = number of bears

L : T = 3 : 2

We can rewrite this as
L/T = 3/2

Cross multiply:
L x 2 = 3 x T

Divide by 3 to get
T = 2/3 L

Since L = 9

T = 2/3 x 9 = 6

In the other ratio we have
T : B = 3 : 4 which we can write as
T/B = 3/4

Cross multiply to get

4T = 3B

B = 4/3 T

Since T = 6, B = 4/3 x 6 = 8

Check
L : T = 9 : 6 = 3: 2 (by dividing both sides of : by 3)
T : B = 6 : 8 = 3:4 (by dividing both sides of : by 2)

Answer:

m = -1

Step-by-step explanation:

may not be accurate, I haven't done this in a while

Answer:

-1y−y1=m(x−x1)

y−6=−1(x+5)

y−6=−1x+(−1×5)

y−6=−1x+−5

y−6=−1x−5

y=−1x−5+6

y=−1x+1

y=−x+1

m=−1

b=1

Step-by-step explanation: Hope this helps!! Mark me brainliest!

D. A statistic is a random variable. A statistic is a figure calculated from a sample, such as the sample mean or the sample standard deviation.

Every statistic is a random variable since samples are chosen at random; these variations cannot be anticipated in advance. It has a mean, a standard deviation, and a probability distribution as a random variable. A statistic's sample distribution is its probability distribution. Generally, sample statistics are calculated to estimate the associated population parameters rather than being ends in themselves. The concepts of mean, standard deviation, and sampling distribution of a sample statistic are introduced in this chapter, with a focus on the sample mean.

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Complete question:

which of the following is true about a sample statistic such as the sample mean or sample proportion?

A. A statistic is constant.

B. A statistic is always known.

C. A statistic is a parameter.

D. A statistic is a random variable.

The range of values for t where both particles are moving in the same direction along the straight line is 1 < t < 2 or t > 2.

A function's derivative is a gauge of how quickly the function is altering at a specific moment. It is that point's tangent line's slope to the function. The limit of the difference quotient as the change in x becomes closer to zero is the definition of the derivative, which is represented by the symbols f'(x) or dy/dx.

Given the displacement of the two particles, the velocity of the given particles can be calculated using the derivative of the distance as follows:

s=t³-4t²+5t

s'(t) = 3t² - 8t + 5 and,

x=t²-4t+4

x'(t) = 2t - 4

Case 1: Both velocities are positive

s'(t) > 0 and x'(t) > 0

3t² - 8t + 5 > 0 and 2t - 4 > 0

t < 1 or t > 2

Case 2: Both velocities are negative

s'(t) < 0 and x'(t) < 0

3t² - 8t + 5 < 0 and 2t - 4 < 0

1 < t < 2

Therefore, the range of values for t where both particles are moving in the same direction along the straight line is 1 < t < 2 or t > 2.

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Answer:

b. 5π/6 is not a solution to cosØ=3√2.

Performing a simulation with 100 trials is a common technique used to assess the impact of chance variation on the results of an experiment or study. The simulation can help you understand how likely it is to see certain results due to chance variation alone, rather than any underlying difference in proportions.

To perform this simulation, you would first need to define the two proportions that you want to compare. For example, you might want to compare the proportion of people who prefer brand A to brand B in a survey.

Next, you would randomly assign each trial to either brand A or brand B based on the defined proportions. For example, if the proportion of people who prefer brand A is 0.6, you would assign 60 out of the 100 trials to brand A and 40 trials to brand B.

After assigning each trial, you would then calculate the difference in proportions between the two groups. This would give you a distribution of differences that you would expect to see due to chance variation alone.

If the observed difference falls within the range of differences expected due to chance variation, you can conclude that the difference in proportions you observed is not statistically significant and may be due to chance.

However, if the observed difference is larger than what you would expect to see due to chance variation, you can conclude that the difference is statistically significant and likely due to an underlying difference in proportions.

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iAnswer:

Step-by-step explanation:

I have no idea

The MAD of the portions of popped balloons is 0.046.

The Mean Absolute Deviation (MAD) is a measure of the average distance between each data point and the mean of the data set. It gives an idea of how spread out the data is around the mean.

To find the mean, median, and MAD (mean absolute deviation) of the given data, we first need to find the average value of the portions of popped balloons.

Mean = (16% + 2/25 + 0.06 + 1/50 + 0.12 + 0.04) / 6

Mean = 0.0975

Therefore, the mean of the portions of popped balloons is 0.0975.

To find the median, we need to first arrange the portions in ascending order:

0.04, 0.06, 1/50, 2/25, 0.12, 16%

Median = (1/50 + 0.06) / 2

Median = 0.035

Therefore, the median of the portions of popped balloons is 0.035.

To find the MAD, we first need to find the absolute deviations of each portion from the mean. We can do this by subtracting the mean from each portion and taking the absolute value:

|0.16 - 0.0975| = 0.0625

|2/25 - 0.0975| = 0.0525

|0.06 - 0.0975| = 0.0375

|1/50 - 0.0975| = 0.0475

|0.12 - 0.0975| = 0.0225

|0.04 - 0.0975| = 0.0575

The MAD is the average of these absolute deviations:

MAD = (0.0625 + 0.0525 + 0.0375 + 0.0475 + 0.0225 + 0.0575) / 6

MAD = 0.046

Therefore, the MAD of the portions of popped balloons is 0.046.

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Every real root of a polynomial with rational coefficients is also a root of a polynomial with integer coefficients.

Suppose that c is a root of the polynomial equation:

[tex]q_n[/tex] × [tex]x^n[/tex] + q_{n-1} × [tex]x^{n-1}[/tex] + ... + q1 × x + q0 = 0

where [tex]q_i[/tex] are rational coefficients. Since c is a root of this polynomial equation, we have:

[tex]q_n[/tex] × [tex]c^n[/tex] + [tex]q_{n-1}[/tex] × [tex]c^{n-1}[/tex] + ... + q1 × c + q0 = 0

Multiplying both sides of the equation by the common denominator of the coefficients [tex]q_i[/tex], we can obtain an equation with integer coefficients. Let d be the least common multiple of the denominators of the coefficients [tex]q_i[/tex]. Then we can write:

d × ([tex]q_n[/tex] × [tex]c^n[/tex] + [tex]q_{n-1}[/tex] × [tex]c^{n-1}[/tex] + ... + q1 × c + q0) = 0

Expanding the left-hand side of the equation, we obtain a polynomial with integer coefficients:

d × [tex]q_n[/tex] × [tex]x^n[/tex] + d × [tex]q_{n-1}[/tex] × [tex]x^{n-1}[/tex] + ... + d × q1 × x + d × q0 = 0

Since c is a root of the original polynomial equation, it is also a root of this polynomial with integer coefficients. Therefore, we have shown that if c is a root of a polynomial with rational coefficients, then it is also a root of a polynomial with integer coefficients.

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Answer: x = 5 and x = 1.

Step-by-step explanation:

To solve fg(x) = gf(x), we need to find the expressions for fg(x) and gf(x) and then set them equal to each other.

fg(x) = f(g(x)) = f(x² + 2) = 2(x² + 2) - 3 = 2x² + 1

gf(x) = g(f(x)) = g(2x - 3) = (2x - 3)² + 2 = 4x² - 12x + 11

Now we set fg(x) equal to gf(x) and solve for x:

2x² + 1 = 4x² - 12x + 11

2x² - 12x + 10 = 0

Dividing both sides by 2 gives:

x² - 6x + 5 = 0

This quadratic equation factors as:

(x - 5)(x - 1) = 0

So the solutions are x = 5 and x = 1.

Therefore, the solutions to fg(x) = gf(x) are x = 5 and x = 1.

x = 4.9

Solution:

We can use the intersecting chords formula:

[tex]7\times7 = 10x[/tex]

[tex]49 = 10x[/tex]

[tex]49\div10=10x\div10[/tex]

[tex]4.9 = x[/tex]

Therefore, x = 4.9.

Answer: 106 127 129 132 135 138 140 158

Step-by-step explanation:

Therefore, the number of students who study only one subject is n(H-only) + n(G-only) = 14 + 28 = 42.

In mathematics, a set is a collection of distinct objects, which are called its elements. These objects can be anything, such as numbers, letters, or even other sets. Sets are usually denoted by capital letters and the elements of a set are listed within braces, separated by.

Given by the question.

i. n (H ∩ G) can be found using the formula:

n(H ∩ G) = n(H) + n(G) - n(H U G)

where n (H U G) represents the number of students who study either History or Government or both.

We know that n(H) = 2x and n(G) = 3x. Also, n(U) = 54, which means the total number of students is 54. Therefore, we can write:

n (H U G) = n(H) + n(G) - n (H ∩ G)

54 = 2x + 3x - x

54 = 4x

x = 13.5

Since x must be a whole number, we can round it up to 14. Therefore, n (H ∩ G) = x = 14.

ii. The number of students who study only one subject can be found by subtracting n (H ∩ G) from n(H) and n(G) respectively, and then adding the number of students who study neither History nor Government.

n(H-only) = n(H) - n (H ∩ G) = 2x - x = x = 14

n(G-only) = n(G) - n (H ∩ G) = 3x - x = 2x = 28

n(Neither) = n(U) - n (H U G) = 54 - 14 = 40

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The horizontal line at $7.25 on the y-axis of the graph is the one with a slope that most accurately depicts the federal minimum wage of $7.25 per hour as of January 1, 2014.

Although it varies from state to state, the federally mandated minimum wage in the United States is $7.25 per hour. The District of Columbia had the highest minimum wage in the US as of January 1, 2023, at 16.50 dollars per hour.

The variable dearness allowance (VDA) component, which takes into account inflationary trends, such as an increase or fall in the Consumer Price Index (CPI), and, if applicable, the housing rent, are included in the computation of the monthly minimum salary.

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Three youngsters are interviewed at random. Population variation is around [tex]18.67[/tex].

The standard deviation refers to the square base of variance, which is the average of the squares departures from the mean. Both metrics capture distributional variability, although they use different measurement units: The units used to indicate standard deviation are the same as the values' original ones.

In statistics, population standard deviation is a crucial indicator of dispersion. further reading. Statisticians compute variance to determine how order to overcome the drawbacks in a data gathering interact to one another. By calculating the population variance, one may also compute the dispersion in relation to the population means.

we need to first find the population mean [tex]u[/tex]

[tex]u = (2 + 4 + 12)/3 = 6[/tex]

To calculate the population variance

[tex]= [(-4)^{2} + (-2)^{2} + 6^{2} ]/3[/tex]

[tex]= (16 + 4 + 36)/3[/tex]

[tex]= 56/3[/tex]

[tex]= 18.67[/tex] (rounded to two decimal places)

Therefore, the population variance is approximately [tex]18.67[/tex].

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The correct option is -C: No, 6 is the geometric mean of 4 and 9, however if the altitude is 6, then the hypotenuse is the geometric mean of the two segments.

An average technique multiplies several values and determines the number's root is known as the geometric mean. You locate the nth root for their product for a collection of n numbers. This descriptive statistic can be used to sum up your data.

Mean Geometric The square root of the product of two numbers is the geometric mean amongst them. The geometric mean of two positive numbers an as well as b is the positive number x as in percentage Cross multiplication results in x² = ab,.

For the given question.

geometric mean of a and b :

From the drawn diagram.

a = 4

b = 9

x = √ab

x = √9*4

x = 6

geometric mean: 6

Applying the altitude rule:

h² = x.y

6² = 9*4

36 = 36

Thus, the geometric mean calculated by friend is correct but the marking on the diagram is wrong.

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Step-by-step explanation:

4)

based on similar triangles and the common ratio for all pairs of corresponding sides we know

LE/LM = LD/LK = DE/EM

because E and D are the midpoints of the longer sides, all of these ratios are 1/2.

1/2 = DE/8

8/2 = 4 = DE

5)

same principle as for 4)

BQ/BA = BR/BC = QR/AC

again, Q and R are the midpoints, so all these ratios are 1/2.

1/2 = QR/10

QR = 10/2 = 5

After answering the provided question, we can conclude that  This expression process frequently results in bolt diameters that are normally distributed, with minor random variations around the target diameter.

In mathematics, an expression is a collection of symbols, digits, and companies that portray a statistical correlation or formula. An expression can be a single number, a mutable, or a combination of both of them. Addition, subtraction, proliferation, division, and exponentiation are examples of mathematical operators.  Expressions are used extensively in mathematics, including arithmetic, calculus, and geometry. They are used in mathematical formula representation, equation solution, and mathematical relationship simplification.

a) Because of the central limit theorem, the volume of soft drinks in cans is likely to follow a normal distribution. Because soft drink manufacturers typically produce a large number of cans filled with the same volume of liquid, the sample size is large.

b) The diameter of bolts immediately after manufacture is also likely to be distributed normally. Bolt manufacturing typically involves a large number of measurements and adjustments to ensure that the bolts are manufactured to exact specifications. This process frequently results in bolt diameters that are normally distributed, with minor random variations around the target diameter.

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Therefore , the solution of the given problem of unitary method comes out to be  the attraction sold 943 child tickets and 736 adult tickets on that particular day.

It is possible to accomplish the objective by using previously recognized variables, this common convenience, or all essential components from a prior malleable study that adhered to a specific methodology. If the expression assertion result occurs, it will be able to get in touch with the entity again; if it does not, both crucial systems will undoubtedly miss the statement.

Here,

Assume the attraction sold x tickets for adults and y tickets for kids.

Based on the supplied data, we can construct the following two equations:

=>  x + y = 1679 (equation 1, representing the total number of tickets sold)

=>  35x + 20y = 44620 (equation 2, representing the total revenue generated)

Using the elimination technique, we can find the values of x and y.

When we divide equation 1 by 20, we obtain:

=>  20x + 20y = 33580 (equation 3)

Equation 3 is obtained by subtracting equation 2 to yield:

=>  15x = 11040

=>  x = 736

When we enter x = 736 into equation 1, we obtain:

=>  736 + y = 1679

=> y = 943

As a result, the attraction sold 943 child tickets and 736 adult tickets on that particular day.

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Consider the line segment AB shown the locations for point C makes ABC a right triangle with hypotenuse AB

In a coordinate pair the points are as represented as (x, y).

The point that forms the right triangle is located by tracing the point on the x axis of of the point A and the point on the y axis of the point B.  This is done below

therefore we can say that the point C that forms the right triangle is

C (2, 3)

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The equation for a cosecant function with vertical asymptotes found at x equals pi over 2 plus pi over 2 times n, where n is an integer, is [tex]f(x) = csc(x - \pi/2)[/tex] .

What is the cosecant function ?

The cosecant function is a trigonometric function that is defined as the reciprocal of the sine function. It is denoted as csc(x) and is defined for all values of x except where sin(x) is equal to zero. The graph of the cosecant function shows a series of vertical lines where the function is undefined, called vertical asymptotes. The value of the cosecant function oscillates between positive and negative infinity as it approaches these asymptotes. The cosecant function is used in trigonometry and calculus to model periodic phenomena such as sound and light waves.

Determining the equation for a cosecant function with vertical asymptotes :

The cosecant function has vertical asymptotes at the zeros of the sine function, which are given by

[tex]x = \pi/2 + n\times\pi[/tex], where n is an integer.

To shift the graph of the cosecant function horizontally by [tex]\pi/2[/tex] units to the right, we subtract [tex]\pi/2[/tex] from the input variable x, so the equation becomes [tex]f(x) = csc(x - \pi/2)[/tex].
[tex]f(x) = csc(x - \pi/2)[/tex] is the equation for a cosecant function with vertical asymptotes found at [tex]x = \pi/2 + n\pi[/tex], where n is an integer.

[tex]g(x) = 4csc(2x)[/tex] is the equation for a cosecant function with period pi, amplitude 4, and vertical asymptotes found at [tex]x = \pi/2 + n\pi[/tex], where n is an integer.

[tex]h(x) = 4csc(3x)[/tex] is the equation for a cosecant function with period [tex]2\pi/3[/tex], amplitude 4, and vertical asymptotes found at [tex]x = \pi/6 + n\pi,[/tex] where n is an integer.

[tex]j(x) = 2csc(x/2)[/tex] is the equation for a cosecant function with period 4pi, amplitude 2, and vertical asymptotes found at [tex]x = 2n\pi[/tex], where n is an integer.

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