my question was why the general solution of the linear second order homogeneous equation is represented as combination of two particular solutions

Answer:

x = -2

y = -5

Step-by-step explanation:

5x - y = -5

3x - 6y = 24

First, multiply the first equation by -6

-30x + 6y = 30

3x - 6y = 24

-27x = 54

x = -2

Now, put in -2 for x and solve for y

5(-2) - y = -5

-10 - y = -5

-y = 5

y = -5

Let's Check

5(-2) -(-5) = -5

-10 + 5 = -5

-5 = -5

So, x = -2   y = -5  is the correct answer.

The speed of the bicycle is 381 meters per minutes

we know that,

revolutions per minute = speed in meters per minute / circumference in meters.

Diameter = 32 inches

Circumference of the wheel = 2πr

r = Diameter/2 = 32/2 = 16

Circumference = 2 × π × 16

=  100.5714 inches

Approximately = 100 inches

we know that,

1 inch = 0.0254 meter

100 inches = x

x = 100 × 0.0254 meters

x = 2.54 meters

Hence:

Revolutions per minute = Speed in meters per minute / circumference in meters.

Speed = Revolutions per minute × circumference in meters.

Speed = 150 RPM × 2.54 meters

Speed = 381  meters per minutes

so, the speed of the bicycle is 381 meters per minutes.

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The numeric value of the greatest sum that can be obtained by Greg is given as follows:

1173.

When two n digit numbers are added, the numbers in the equivalent positions are added with each other, that is, starting from the end of the number to the begining.

For example, 21 + 32:

Hence the sum is of 54.

Hence the greatest sum would be obtained as follows:

Hence the sum would be given as follows:

642 + 531 = 1173.

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The value of 3h(2) + 4g(1) is 39

You can simply replace those variables in function with the value you know of them and then operate on those values to get a final value. This is the result of that expression at those values of the considered variables.

Given;

h(x)= 3x        x<2

     = 4x+1     x>2

g(x)= x^2+2   x<3

     = x^3        x>3

Now, substituting the values;

3h(2)= 3*(4*2+1)

=3*9

=27

4g(1)= 4(1^2 + 2)

=4(3)

=12

3h(2)+4g(1)=27+12

=39

Therefore, the answer of the function will be 39

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there exists an integer that is greater than 5 and whose square is smaller than 40, express as a mathematical quantifiers ∃x∈ℤ: x > 5 ∧ x² < 40

1.  Start by writing down the equation: x² < 40

2.  Take the square root of both sides of the equation to isolate the variable x: √x² < √40

3.  Simplify both sides of the equation: x < √40

4.  Check that the result makes sense: x < 6.3

5.  Take the integer part of the result: x < 6

6.  Since the requirement is that x is greater than 5, the answer is x = 6.

We start by writing the equation x² < 40 and then take the square root of both sides to get x < √40. We check that the result is reasonable and then take the integer part of the result to get x < 6. Since the requirement is that x is greater than 5, the answer is x = 6. To confirm, we can calculate that x² = 36 which is smaller than 40, and indeed, 6 is greater than 5. Therefore, the integer that is greater than 5 and whose square is smaller than 40 is 6. This can be expressed mathematically as ∃x∈ℤ: x > 5 ∧ x² < 40 which translates to: There exists an integer x such that x is greater than 5 and the square of x is smaller than 40.

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The total cost of materials for one candle is $2 and she pays $40 for the booth for the day.

Therefore, the total cost for making and selling one candle is $2 + $12 + $40 = $54

To cover all her expenses, Leila must sell enough candles to make $54 from each candle.

She must sell 54/$12 = 4.5 candles

Therefore, Leila needs to sell at least 5 candles to cover all her expenses.

At that point, her sales and expenses will be 5 candles x $12/candle = $60

Therefore, Leila's sales and expenses will be $60 at that point.

The constants a and b can be found by ensuring the function is continuous and that the limits at each point match. This can be done by setting the values of a and b so that the function and its derivative are equal at each point.

To find the constants a and b for the given piece-wise cubic function, we must consider the definition of continuity and limits. Firstly, for the function to be continuous, the function and its derivative must be equal at each point. Thus, we need to set the values of a and b so that the function and its derivative are equal at the points x=1, x=2 and x=3. We can use this to create a system of equations to solve for a and b. Once we have determined the values of a and b, we can then check that the limits at each point match. This can be done by computing the left and right hand limits at each point, and ensuring they are equal. If the limits are equal, then the function is continuous and the constants a and b are correct.

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Use the intercepts to graph the equation y = 18-2x

What is meant by graph ?

Interpret What real-life quantities do the x- and y-intercepts

represent in this situation:

when x = 0, y = 18,  when y = 0 x = 9

The x intercept is 9,  y intercept is 18.

After 9 hours the gas is run not .

The amount of gas in a car when it is not running

18 × 1/4 = 4.5

4.5 = 18-2x

x= 6.75 hours

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The reflection of point G across the x-axis would be at (-3, 2).

A point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.

The coordinates of point G in the question are (-3,-2)

When a point is reflected across the x-axis, the x-coordinate remains the same, but the sign of the y-coordinate is changed.

Therefore, the reflection of point G, which is located at (-3, -2), across the x-axis would be at (-3, 2).

The x-coordinate is the same, but the y-coordinate has changed from negative to positive.

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The question seems incomplete, the completed question would be as:

The coordinates of point G are (-3,-2)

What would be the reflection point G across the x-axis?

Analyze the diagram of the trapezoid gives BA = 10.5 units

A trapezoid (also known as a trapezium) is a four-sided geometric shape that has two parallel sides (the bases) and two non-parallel sides (the legs). The bases of a trapezoid can have different lengths, but the legs must be congruent and parallel to each other.

Let's analyze the diagram below and complete the instructions:

Since CB / CA = DE / DF

where CB = 6, DE = 4, EF = 7

We can say:

DF = DE + EF = 4 + 7 = 11

CA = CB + BA = 6 + BA

CB / CA = DE / DF

6/(6 + BA) = 4/11

6 × 11 = 4(6 + BA)

66 = 24 + 4BA

24 + 4BA = 66

4BA = 66 - 24

4BA = 42

BA = 42/4

BA = 10.5 units

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Answer: [tex]2\sqrt{13}-\sqrt{697}[/tex]

Step-by-step explanation:

Since ABD is a right triangle, 14^2 = 12^2 + BD^2

196 = 144 + BD^2

52 = BD^2

BD = [tex]\sqrt{52} = 2\sqrt{13}[/tex]

ADC is also a right triangle, so 29^2 = 12^2 + DC^2

841 = 144 + DC^2

DC^2 = 697

DC = [tex]\sqrt{697}[/tex]

Since x = BD + DC,

x = [tex]2\sqrt{13} +\sqrt{697}[/tex] :>

The measures of the angles are Z = 146 degrees, X and W = 34 degrees and B = 119 degrees

Question 4

From the question, we have

<Y = 146 degrees

Because the lines XY and WZ are equal, it means that

<Z = 146

So, we have

X = W = (180 - 146)

Evaluate

X = W = 34

Question 6

Here, we have

Sum of the angles = 180 degrees

This gives

9x + 2 + 5x - 4 = 180

Evaluate the like terms

14x = 182

Divide by 14

x = 13

So, we have

B = 9 * 13 + 2

B = 119 degrees

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We have to find solutions for the given differential equation when

y = 0,  y = 1/(k-x) and y(a) = b using ordinary differential equations.

We have dy / dx = y^2

a. let's define the given y = 1/(k-x)

derive the above equation with respect to x :

                                  dy / dx = y' = 1/(k-x)^2

So this is the general solution.

b) We would want a singular solution that is not included in the general form. Here we could use y = 0

dy / dx = 0 = 0^2

so,this is a trivial singular solution.

c) If we have y(a) = b,

then, y = 1(k-a) = b

To find the value of k we use the other condition: y'(^2) = 4 x y(2)

           1/(k-a)^2 = 4 x  1/(k-a)

              k - 2 = 1/4

              therefore, k = 9/4

hence we got, y = 1/(9/4 - a) = b

So the points (a, b) are of the form: (a, 1/(9/4 - a))

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The complete question is :

Find a general solution of the differential equation dy/dx=y2

b. Find a singular solution that is not included in the general solution.

c. Inspect a sketch of typical solution curves to determine the points (a,b) for which the initial value problem (y′)2=4y,y(a)=b

The sampling used in this situation is Stratified sampling
In the question, students are selected from subgroups of Sophomore, Junior, and Senior classes.

We are told that 49, 34, and 48 students are selected from the Sophomore, Junior, and Senior classes with 496, 348, and 481 students respectively.

This means that the sample numbers of 49, 34 and 48 students were selected in proportion to the subgroup sizes of 496, 348, and 481 students respectively.

Thus, due to the fact that subgroups were selected & that sample number of students were also selected in proportion to their respective subgroup sizes.

This is, therefore, a stratified sampling.

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Q: 49, 34, and 48 students are selected from the Sophomore, Junior, and Senior classes with 496, 348, and 481 students respectively. Identify which type of sampling is used and why?

Answer:

it is easy to add all the numbers up then you will get 77 then tell me how many number should you add to get to 100 it is easy trust me

Step-by-step explanation:

As per the cone surface, the general direction in which the shatter cone ridges converge is south.

The term cone in math is defined as a three-dimensional shape in geometry that narrows smoothly from a flat base  to a point called the apex or vertex.

Here we have given that Shatter cones are key features that uniquely characterize impact structures

In this one their presence provides evidence that an impact occurred. Here we also know that the Shatter cone surfaces exhibit a texture with a series of intersecting ridges that produce its conical geometry.

Based on the definition of cone surface we have identified that the the correct  option is (a) that is the south.

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The amount of the principal investment is, $1200. So option A is correct

Simple interest is an amount which is calculated for a certain period of time and for a certain rate, in simple interest calculation there are three factors which we consider: principal amount(P), rate(R) and time(T).

Every time we calculate simple interest the principal values will be the same unlike compound interest.

Formula;    Simple Interest(I) =  P × R × T/100

A = P(1 + RT)

Given that,

A = $1422

R =  3.7%

T =  5 year

P  = ?

After 5 year total amount

1422 = P ( 1 + 0.037 × 5)

P  =   1422/( 1 + 0.037 × 5)

   =    1422/1.185

P  =    1200

Hence, the principal amount is $1200

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a) The probability of a student living within walking distance of campus is 0.1521.

To find the probability of a student living within walking distance of the campus, we can use the formula for conditional probability:

P(A and B) = P(A) * P(B|A).

For this situation, An is "living within walking distance to campus" and B is "living with parents."

Thus, P(A and B) = (0.23)(0.67) + (0.06)(0.33) = 0.1521.

Hence, the probability of a student living within walking distance of campus is 0.1521.

b)  The probability given a student living within walking distance of campus they experience with their parents is 0.67.

To find the probability given a student day to day routines within walking distance to campus they experience with their parents, we can use the formula for conditional probability:

P(A|B) = P(A and B)/P(B).

In this situation, A is "living with parents" and B is "living within walking distance to campus."

Thus, P(A|B) = P(A and B)/P(B) = (0.23)(0.67)/0.1521 = 0.67.

Subsequently, the probability gave a student living within walking distance of campus experience with their parents is 0.67.

c) Living with parents and living within walking distance are independent events.

To decide whether living with parents and living within walking distance are independent events, we can analyze P(A|B) with P(A).

On the off chance that P(A|B) = P(A), the events are independent. For this situation, P(A|B) = 0.67 and P(A) = 0.67, so living with parents and living within walking distance are independent events.

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Answer: 3 + y = 24

Step-by-step explanation: Hello!

If she rode 3 miles one day and y miles the other 6 days of the week, and in total that adds up to 24 miles, that means the equation should be addition.

3 miles + y miles = 24 miles
Monday, other 6 days, total seven days

The equation should be:
3 + y = 24

Answer:

a)  Domain:  (-∞, 10)

b)  k⁻¹(8) = 6

c)  CBD

d)  Range:  (-∞, 7)

e)  k⁻¹(0) = 4

f)  x = -5

Step-by-step explanation:

The domain of the inverse of a function is the same as the range of the original function.  

Therefore, the domain of the inverse function is:

The inverse of a function is its reflection in the line y = x.

Therefore, given that function k(6) = 8 then:

The inverse of a function is its reflection in the line y = x.

We cannot find k⁻¹(-9) since we have not been given the value of x when k(x) = -9.

The range of the inverse of a function is the same as the domain of the original function.  

Therefore, the range of the inverse function is:

The inverse of a function is its reflection in the line y = x.

Therefore, given that function k(4) = 0 then:

The zero of the inverse function is the y-intercept of the original function.  Therefore, given that k(0) = -5 then:

Therefore, x = -5 is a zero of k⁻¹(x).

The equation of the exponential function is y = 0.2⋅2^x

The steps for this process are as follows:

Step 1: Set up the equation.

The equation of an exponential function has the form y = a⋅b^x

where a and b are constants and b is greater than 0 and not equal to 1.

Step 2: Find the constants.

We can use the given points to find the constants in the equation. For example, when x = 0, y = 0.2.

So we can set 0.2 = a⋅b^0.

This means that a = 0.2 and b^0 = 1,

so b = 1.

Step 3: Write the equation.

Substituting a = 0.2 and b = 1 into the equation gives us y = 0.2⋅2^x.

This is the equation of the exponential function represented by the table.

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Answer:

1.7320508075689

Length of malcom's story is 418lines, the operation used to solve this problem is; lines of zac story+ extra lines of malcom story.

Usually, simplification involves proceeding with the pending operations in the expression. Like, 5 + 2 is an expression whose simplified form can be obtained by doing the pending addition, which results in 7 as its simplified form. Simplification usually involves making the expression simple and easy to use later.

Given;

Number of lines in zac story=273

Number of lines in malcom story= 273+145

Now.

Total number of lines in malcom story

=273+145

=418

Therefore, the answer after simplification will be 418

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Arithmetic mean deals will middle number while geometric mean deals with typical value

From the question, we have

Arithmetic mean and Geometric mean

The arithmetic mean is the sum of a set of numbers divided by the number of items in the set, and is often used to find the "middle" value of a dataset.

On the other hand, the geometric mean is the nth root of the product of a set of numbers and is often used to find the "typical" value of a dataset when dealing with quantities that can increase or decrease exponentially.

The above definitions imply that the difference between the arithmetic mean and geometric mean can be significant when dealing with sets of numbers that have large variations in values.

In general, the geometric mean will be less than or equal to the arithmetic mean.

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The graph that represents an exponential decay function is the second graph.

For you to determine which of the graphs represents an exponential decay, we will need to check the curve that passes through or lies on the x-axis.

we can draw a graph by putting the value of x and we are getting the value of y in this way we get the coordinates of a point

From both figures, we can see that the curve of the second graph almost lies on the x-axis as it curves downwards. This makes it an exponential decay function.

Hence the graph that represents an exponential decay function is the second graph.

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complete question is

Which description matches the function represented by this graph?

A. exponential growth

B. linear growth

C. linear decay

D. exponential decay

Answer:

B or 2

Step-by-step explanation:

The answer above is correct.

The sample has 3.33% of defective bulbs so, B. The shipment will not be replaced.

A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.

Given, If a shipment has at least 5% defective light bulbs, it will be replaced.

Now, A sample of 60 light bulbs has 2 that are defective.

Therefore, The percentage of defective bulbs is,

= (2/60)×100%.

= (1/30)×100%.

= 3.33%, So, the shipment will not be replaced.

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Applying the linear pair definition, m<KGH = 60°.

In geometry, a linear pair is a pair of adjacent angles that are supplementary, meaning that their measures add up to 180 degrees. The two angles must be adjacent, meaning that they share a common side or vertex, and they must be linear, meaning that they are formed by two intersecting lines.

Therefore:

m<FGK + m<KGH = 180

Substitute

120 + m<KGH = 180

m<KGH = 180 - 120

m<KGH = 60°

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The linear equation for the cost of lemonades is 1.50*x+6 where x=5 i.e. A   and he spends $54.75 in total.

What is a linear equation ?

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form

Ax + B = 0

e.g. x-10=0. Here, x is a variable, A is a coefficient and B is constant.

The standard form of a linear equation in two variables is of the form

Ax + By = C

e.g. 2x-4y=10. Here, x and y are variables, A and B are coefficients and C is a constant.

Now,

Cost of lemonade=$1.5, number of lemonades=5

Cost of hamburger=$8.25, number of hamburgers=5

Tip=$6

Hence,

        Total cost=5*1.5+5*8.25+6

          Total cost =$54.75

Cost of lemonades only = 5*1.5+6

In type of linear equation  1.50*x+6 where x=5

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