solve this system of equations by using the elimination method.
2x+y=4 x+3y=2

Answer:

the answer is B (0, -2)

Step-by-step explanation:

To find the point on the curve y = -x^2 - 2 that is closest to the point (0,1), we can use the method of least squares.

The method of least squares involves finding the point on the curve that minimizes the square of the distance between the point and the curve.

To begin, we can define a function d(x) = (y - (-x^2 - 2))^2 which represents the square of the distance between the point (x,y) on the curve and the point (0,1)

then we need to find the derivative of d(x) and set it to zero to find the minimum,

d'(x) = 2(y + x^2 + 2 - 1) = 2(y + x^2 + 1)

by setting d'(x) = 0 we get y = -x^2 - 1,

so the point on the curve that is closest to (0,1) is (0, -1) and the answer is B (0, -2)

Note that the point (0,-1) is on the curve y = -x^2 - 2 but not the point on the closest.

The solution of the quadratic equation is obtained as -1 ± √7. Option A and B.

We know that a quadratic equation has to do with the kind of equation that have the general form; y = ax^2 + bx + c. This implies that we are going to have the highest power of the variable to be x.

As such, we have the equation that have been written as; x^2 + 2x – 6 = 0. We are to find the roots of the quadratic equation which are the values for x. When we solve the quadratic equation we have;

-1 ± √7

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x is the probability of people who receive their original hats back out of the n people. All possible re-assignments are equally likely.

x is the number of people out of the n people who are lucky enough to receive their original hats back from the hat-check attendant. All possible re-assignments of hats to people are equally likely. This means that each person has an equal chance of getting their own hat back, and the probability of any particular person getting their own hat back is 1/n. The expected value of x is therefore n/2, since the average of all possible outcomes is the expected value. The variance of x is (n-1)/12, since the variance is the average of the squared differences from the expected value.

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This method of observing the behaviour of each group of children is best characterized as experimental.

Experimental research is a study worked with a scientific approach using two sets of variables. The first set works as a constant, which we use to measure the differences of the second set.

This can be regarded as something that depends on other factors, such as a test score, which can be considered a dependent variable because it could change depending on several factors, such as how much you studied.

It should be noted that dependent and independent variables are variables in mathematical modelling and experimental sciences; however, the dependent variables can be seen as ones whose values are studied under the supposition.

Hence the correct option is a.

--The given question is incomplete; the complete question is

"A researcher randomly assigned boys and girls to each of two groups. One group watched a violent television program, while the other watched a nonviolent one. The children were then observed during free play, and the incidence of aggressive behaviour was recorded for each group. This method is best characterized as"--

a. Experimental

b. Case study

c. Correlational

d. Observation"--

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To lose the least amount of money in the sales promotion, you should order the minimum number of hats and pairs of sunglasses while still satisfying the constraints given.

Let x be the number of hats you order and y be the number of pairs of sunglasses you order.

From the information given, we can set up the following constraints:

y = 2x (because you will sell at least twice as many hats as pairs of sunglasses)

y <= 100 (because your supplier tells you that your order of sunglasses cannot exceed 100 pairs)

x + y >= 210 (because you estimate that you should order at least 210 items in all)

We also know that you will lose $3 on every hat and $2 on every pair of sunglasses sold. So the objective is to minimize the cost:

Cost = 3x + 2y

Now we can use the constraints to set up a linear optimization problem. We can use a solver to minimize the cost and find the optimal values of x and y that meet the constraints.

Solving the problem gives us x = 140, and y = 280, so you should order 140 hats and 280 pairs of sunglasses to lose the least amount of money in the sales promotion.

PART 1: The solution of the inequality is x < 3.5 and the second graph shows the solution.

PART 2: The inequality she solved is (2/5)x - 4/5  ≤ 1 1/5.

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value e.g. 5 < 6, x ≥ 2, etc.

PART 1

First, solve for x in 9.5x -1.25 < 32

9.5x -1.25 < 32

9.5x < 32 + 1.25

9.5x < 33.25

x < 33.25/9.5

x < 3.5

Since x < 3.5, the circle of the graph will not be shaded. Thus, the second graph is the graph of x < 3.5.

PART 2

The graph of the inequality shown there is x ≤ 5. Thus, any inequality which when solved will give x ≤ 5 will be the inequality that represent the graph.

Let's try (1/5)x - 3/5  ≤ 1/5 and see:

(1/5)x - 3/5  ≤ 1/5

(1/5)x ≤ 1/5 + 3/5

(1/5)x ≤  4/5

x ≤ 4

This is not the inequality

Let's try (2/5)x - 4/5  ≤ 1 1/5:

(2/5)x - 4/5  ≤ 1 1/5

(2/5)x - 4/5 ≤ 6/5

(2/5)x ≤ 6/5 + 4/5

(2/5)x ≤ 2

2x ≤ 10

x ≤ 5

This is the inequality

Thus, she solved (2/5)x - 4/5  ≤ 1 1/5

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There are 15120 integers greater than 5400 that have distinct digits and do not contain 2 and 7.

The number of required integers is the number of integers greater than 5400 which have distinct digits and do not contain 2 and 7. This can be calculated with the formula:

Number of Required Integers = (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) - (2 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) = 30240 - 15120 = 15120.

This is because for the first part of the equation, we must calculate the total number of possible permutations of distinct digits from 0-9 (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1). For the second part, we must calculate the number of permutations of digits from 0-9 excluding 2 and 7 (2 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1). We then subtract the second part from the first to get the required answer.

Therefore, there are 15120 integers greater than 5400 that have distinct digits and do not contain 2 and 7.

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Answer: 1/2n = 1/4n - 3n+3/2n

Start by multiplying both sides by 2n to get rid of the fractions:

1 = 1/2 - 3n+3/n

Then by multiplying both sides by n to get rid of the fraction on the right side:

n = -3n+3

Then by adding 3n to both sides:

4n = 3

Then by dividing both sides by 4:

n = 3/4

There is no extraneous solutions in this case, because no denominators are equal to zero and all the operations are valid. Therefore, the solution is n = 3/4

Step-by-step explanation:

Carmen scored 50%of the total number of items in second test, the number of items in the second test is, 75

The percentage is a mathematical term, which means a number or a ratio which is represented in fractions of 100. We use the '%' symbol to represent the percentage.

Formula for percentage;

Percentage = (present quantity/whole quantity)×100

Given that,

In the first Summative Test,

Carmen scored 20 points which is 80% of the number of items

In the second test,

Carmen scored 30 points,

Her total score is 50% of the number of items

The total number of the items in the second test = ?

Suppose, total number of items in the first test = x

                                               in the second test = y

⇒  20 points =  80x/100

⇒  20 points =  0.80x

⇒  x   =  25

⇒  0.5(x + y)  = 20 + 50

⇒     x + y = 100

⇒     y = 75

Hence, the total number of items in the second test is 75

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The graphical summary of salaries is an example of b. descriptive statistics.

Descriptive statistics is the branch of statistics that is used to summarize, organize, and describe data. It is used to present data in a clear and meaningful way, so that patterns, relationships, and trends can be easily understood. A graphical summary of all employees' salaries, such as a histogram or a bar chart, is an example of descriptive statistics.

This graphical summary represents the entire population of employees and not a subset of it, so it's not a sample. The factory owner is not making any inferences about the population based on the data, so it's not an example of statistical inference. And this is not an experiment because no manipulation of the data was made.

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The arrangement with descending trend is 1.247,1.244,1.234

A decimal number is a number that consists of a whole number and a fractional part separated by a point. For example – 3.5, 6.79, 78.32, etc.

Given here: The decimal numbers 1.234,1.244,1.247

Now  If one decimal in the list has a higher whole number than the others, it’s automatically the greatest number. If a decimal in the list has a smaller whole number than the others on the list, it’s automatically the smallest number. . Create enough rows and columns for all of the digits and number spaces (including the decimal point). Going from left to right, label the columns as tens, ones, decimal, tenths, hundredths, and thousandths.

Thus,

 ones  decimals tenths Hundreths thousands

   1           .              2           3                 4

   1           .              2           4                 4

   1           .              2           4                  7

∴ The order from greatest to least is 1.247,1.244,1.234

Hence, The arrangement with the descending trend is 1.247,1.244,1.234

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Answer:

40%

Step-by-step explanation:

To find the percent discount, take the original price and subtract the discounted price.

21 - 12.60=8.4

Divide this by the original price.

8.4/21 = .4

Change this to percent form.

.4 = 40%

Answer:

8.4

Step-by-step explanation:

The price of a bag = 21

Magan paid 12.60

00. 21

12.00

- - - - - - - - -

8.4

The amount in the account after 15 years is, $6,671.67

A loan or deposit amount is subject to compound interest as interest. In everyday life, it is the most prevalent idea. Both the principal and the interest accrued over time affect the compound interest for a given amount. Between compound and simple interest, this is the primary distinction.

[tex]A = P[1 + r/n]^{nt}[/tex]

A  = P + I

Where,

A = compound interest

P = Principle Amount

r = Interest rate

n = number of period for which interest is calculated

t = time

Given that,

A equal to P quantity 1 plus 0.054 over 2 end quantity all raised to the power of 2 times t

an annual interest rate of 5.4% compounded semiannually

the initial investment P is $3,000

the amount in the account after 15 years = ?

[tex]A = 3000[1 + 0.054/2]^{2t}[/tex]

after 15 years compound interest,

[tex]A = 3000[1 + 0.054/2]^{30}[/tex]

A = $6,671.67

A = P + I

P (principal) = $3,000.00

I (interest) = $3,671.67

Hence, the amount in the account after 15 years is, $6,671.67

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The volume of the common region between two spheres, each with radius r, is 4/3π(r^3 - 32r^6).

The volume common to two spheres, each with radius r, is given by the equation V = 4/3πr3.

The volume of the common region between two spheres, each with radius r, is given by the formula:

V = 4/3π(r^3 - (r^2 - d^2)^3/2),

where d is the distance between the centers of the spheres.

In this case, the distance between the centers is 2r, since the center of each sphere lies on the surface of the other sphere.

Thus, the volume of the common region is 4/3π(r^3 - (r^2 - (2r)^2)^3/2)

= 4/3π(r^3 - (r^2 - 4r^2)^3/2)

= 4/3π(r^3 - (r^2 - 4r^2)^3/2)

= 4/3π(r^3 - (4r^2)^3/2)

= 4/3π(r^3 - 64r^6/2)

= 4/3π(r^3 - 32r^6)

= 4/3π(r^3 - 32r^6)

= 4/3π(r^3 - 32r^6).

Therefore, the volume of the common region between two spheres, each with radius r, is 4/3π(r^3 - 32r^6).

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Complete question : What is the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere?

So, the solution is x = 1, y = -1, z = 3.

One way to solve the system is using Gaussian elimination with back-substitution. The steps are:

1. Rewrite the system in matrix form, called the augmented matrix:

[3 3 6 | 9]

[1 1 2 | 3]

[2 5 10| 15]

[-1 2 4 | 6]

2. Perform row operations to convert the matrix to an upper triangular matrix.

[1 1 2 | 3]

[0 2 4 | 3]

[0 2 6 | 6]

[0 0 -2 | -6]

3. Use back-substitution to find the values of the variables. Starting with the last equation:

-2z = -6 => z = 3

4y + 6(3) = 6 => y = -1

x + 2(-1) + 2(3) = 3 => x = 1

So, the solution is x = 1, y = -1, z = 3.

Another way is Gauss-Jordan elimination, it's similar with Gaussian elimination but it will change the matrix to a row echelon matrix and then to a reduced row echelon matrix.

So, the solution is x = 1, y = -1, z = 3.

Complete Question: solve the system using either gaussian elimination with back-substitution or gauss-jordan elimination. (if there is no solution, enter no solution. if the system has an infinite number of solutions, express x, y, and z in terms of the parameter t.)                                                                                                              3x + 3y + 6z = 9

x + y + 2z = 3

2x + 5y + 10z = 15

−x + 2y + 4z = 6

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The probability of tossing "heads" on each of your first two tosses is 0.25

Probability = n(E)/n(S)

Where E is the possibility of an event occurring

and S is the sample space i.e., the total number of possible outcomes.

Probability is the measure of how likely an event can happen. The probability of any event lies between 0 and 1 only.

The sample space for when you toss one coin is - H, T

So the probability to get an H is= 1/2

=0.5

Similarly, The sample space for when you toss two coins is HH, TT, HT, TH

So, the probability to get an H on each of your first two tosses is = 1/4

=0.25

Therefore, The probability of tossing "heads" on each of your first two tosses is 0.25

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If Canada specializes in producing wheat and the US specializes in producing TV sets, both countries will benefit from trade.

To determine this if Canada produces wheat, the opportunity cost is 1/2 TV set. If the US produces wheat, the opportunity cost is 2 TV sets. So, Canada has a comparative advantage in producing wheat. Similarly, the US has a comparative advantage in producing TV sets.

Let's say Canada produces 100 bushels of wheat, and the US produces 100 TV sets. If they trade with each other, Canada can exchange 50 bushels of wheat for 50 TV sets, and the US can exchange 50 TV sets for 50 bushels of wheat. Both countries would benefit from this trade.

Canada would end up with 50 TV sets and 50 bushels of wheat, while the US would end up with 50 bushels of wheat and 50 TV sets. Both countries would be better off than if they had not traded.

Opportunity cost in mathematics is a term that is used to describe the cost of one alternative in terms of the benefits forgone by the best alternative. It can also be represented in terms of mathematical equations.

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The height of the tree is 39 ft from the ground.

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse.

Let us suppose the height of the tree = l

The distance from the point on the ground to the top of the tree is thus:

h = 11 + 2(l)

Given that the base = 80 ft.

To find the height of the tree we can use the Pythagorean theorem.

The Pythagorean theorem is given as follows:

a^2 + b^2 = c^2

(80)^2 + (l)^2 = (11 + 2l)^2

Using the identity (a + b) ^2 = a^2 + 2ab + b^2 we can write the equation as:

6400 + l^2 = 121 + 44l + 4l^2

Take all the terms on a single side of the equation:

4l^2 - l^2 + 44l + 121 - 6400= 0

3l^2 + 44l - 6279 = 0

Solve the following quadratic equation using the equation of the quadratic function:

l = 39

Hence, the height of the tree is 39 feet.

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The solutions that match the given differential equation are a)y=0 and b)y=2x.

The differential equation is a homogeneous linear differential equation with constant coefficients, which can be written in the form of y" + p(x)y' + q(x)y = 0. The general solution to this type of equation is y = c1e^(rx) + c2e^(rx) where r is the root of the characteristic equation r^2 + p(x)r + q(x) = 0.

In this case, the equation is of the form xy'' - y' = 0. By dividing both sides by x, we get y'' - (1/x)y' = 0, which is a homogeneous linear differential equation with constant coefficients. The characteristic equation is r^2 - (1/x)r = 0. The roots of this equation are r1 = 0 and r2 = 1/x.

Therefore, the general solution to this differential equation is y = c1 + c2x.

y=0 is a solution of the differential equation since it satisfies the equation when plugged in.

y=2x is also a solution of the differential equation since it also satisfies the equation when plugged in.

y=2x^2 and y=2 are not solutions of the differential equation because when plugged into the equation they don't satisfy it.

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The green triangle has an area of 6 sq units. The purple triangle has an area of 6 sq units. If the orange triangle is placed directly next to the green triangle such that the two triangles together make a rectangle, the total area of this rectangle is 12 sq units, which is half the area of the green triangle.

We know that the area of a triangle is given by

1/2 X Base X height

Now, for the green triangle, the area  will be

1/2 X (5 - 1) X (9 - 6)

= 1/2 X 4 X 3 sq units

= 6 sq units

The distance between the highest vertex and the base's y coordinates for the purple triangle will make up the height. hence we get

1/2 X (9 - 5) X (4 - 1)

= 1/2 X 4 X 3 sq units

= 6 sq units

Now if the orange triangle is placed on the graph in such a way that it makes a rectangle with the green one, the length and width of the rectangle will be the same as the base and height of the triangle

Hence the area of the rectangle will be

4 X 3 sq units

= 12 sq units

We can see that it is double the area of the purple triangle.

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The probability that one player receives all the 13 hearts is 1/2598960.

How likely something is to occur can be determined using probability. For many events, it is challenging to make an accurate prediction.

The number of ways to distribute the 13 hearts among the 4 players is 4!. The number of ways to distribute the remaining 39 cards among the 4 players is 39!/(13!13!13!13!). Hence, the total number of ways to distribute the 52 cards among the 4 players is (52!)/(13!13!13!13!).

Therefore, the probability that one player receives all the 13 hearts is (4!)/((52!)/(13!13!13!13!)) = 1/((52515049)/(13121110)) = 1/2,598,960.

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Trigonometry is best used in navigation to estimate where to point the compass to get a straight line. It will be simple to pinpoint a location as well as find distance and see the horizon using a compass and trigonometric functions in navigation.

The area of mathematics known as trigonometry examines the correlation between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are used to study this relationship. The concept of trigonometry was developed by the Greek mathematician Hipparchus, and the word trigonometry is a derivative of Latin from the 16th century.

One of the most significant area of mathematic is rigometry. The words "Trigonon" and "Metron," which mean respectively "triangle" and "measure," are combined to form the word "trigonometry." It is the investigation of how a right-angled triangle's sides and angles relate to one another. Therefore, using formulas and identities based on this relationship aids in determining the measure of a right-angled triangle's unknown dimensions.

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Full question:

What is the best use of trigonometry?

The transformation with respect to dilation will only preserve the angle measure i.e. B.

What is dilation in mathematics?

Dilation is a process of changing the size of an object or shape by decreasing or increasing its dimensions by some scaling factors. For example, a circle with radius 10 unit is reduced to a circle of radius 5 unit. The application of this method is used in photography, arts and crafts, to create logos, etc. In Geometry, there are four basic types of transformations. They are:

Some features of shapes that remain unchanged during the dilation transformations are:

Now,

As in dilation Each angle of the figure remains the same.

Hence,

              The transformation with respect to dilation will only preserve the angle measure

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The measure of the other acute angle of the right angled triangle is 54⁰.

For the stated question:

Length of one side = 8 ft.

Let angle A = 36⁰.

For right angled triangle, Angle B = 90⁰

Then,

∠A + ∠B + ∠C = 180⁰

36⁰ + 90⁰ + ∠C = 180⁰

∠C = 180⁰ - 126⁰

∠C = 54⁰

Thus, the other acute angle is found as 54⁰.

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The correct question is-

1. A right triangle has one leg that measures 8ft What is the measure of the acute triangle? Assume that the angle is 36⁰

Answer: To solve the system of equations, we can use the substitution method. We'll start by isolating one of the variables in one of the equations.

The first equation is 3x - 2y = 17

We can solve for x by adding 2y to both sides:

3x = 17 + 2y

x = (17 + 2y)/3

We can substitute this expression for x into the second equation:

5((17 + 2y)/3) + 3y = 5

Simplifying, we get:

50 + 10y + 3y = 5

13y = -45

y = -3.5

Now we can substitute this value of y into the equation we found earlier for x:

x = (17 + 2(-3.5))/3

x = (17 - 7)/3

x = 10/3

x = 3.333

So the solution of the system of equations is (x,y) = (3.333, -3.5)

Which is not in the options provided, so the answer is no solution.

Step-by-step explanation: