Show that 1/x is an integrating factor of the equation dy/dx-y/x = 3x​​

Answers

Answer 1

Suppose we're asked to solve the first order linear differnetial equation,

[tex]\longrightarrow\rm{\dfrac{dy}{dx}+y\cdot P(x)=Q(x)}[/tex]

To solve this equation we have to find out a term called 'Integrating Factor' given by,

[tex]\rm{I_f=e^{\int P(x)\ dx}}[/tex]

such that the solution of the equation is,

[tex]\displaystyle\longrightarrow\rm{y\cdot I_f=\int Q(x)\cdot I_f\ dx}[/tex]

Here the given differential equation is,

[tex]\longrightarrow\rm{\dfrac{dy}{dx}-\dfrac{y}{x}=3x}[/tex]

[tex]\longrightarrow\rm{\dfrac{dy}{dx}+y\left(-\dfrac{1}{x}\right)=(3x)}[/tex]

Here,

[tex]\longrightarrow\rm{P(x)=-\dfrac{1}{x}}[/tex]

Integrating wrt x,

[tex]\displaystyle\longrightarrow\rm{\int P(x)\ dx=-\int\dfrac{1}{x}\ dx}[/tex]

[tex]\displaystyle\longrightarrow\rm{\int P(x)\ dx=-\ln|x|}[/tex]

[We will consider constant of integration as zero.]

Then the integrating factor is,

[tex]\longrightarrow\rm{I_f=e^{\int P(x)\ dx}}[/tex]

[tex]\longrightarrow\rm{I_f=e^{-\ln|x|}}[/tex]

Since [tex]\rm{e^{b\,\ln(a)}=a^b,}[/tex]

[tex]\longrightarrow\rm{I_f=x^{-1}}[/tex]

[tex]\longrightarrow\rm{\underline{\underline{I_f=\dfrac{1}{x}}}}[/tex]

Hence the integrating factor for the given differential equation is 1/x.


Related Questions

if a rectangle is twice as long as it is wide and has a perimeter of 54 meters, what is the area of the rectangle?

Answers

Answer:

l = 2w   (length is twice the width)

P = 2l + 2w

54 = 2l + 2w  (but we know l equals 2w so we can replace l with 2w)

54 =2(2w) + 2w

54 = 6w

w = 9

l =2(9)   (l = 2w

l=18

A = lw

How many zeros can the polynomial 2x² 3x 4 have?

Answers

There are two zeros of the polynomial 2x²-3x+4.

What is Polynomial?

Polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables

The given polynomial is 2x²-3x+4.

Two times of x square minus three times of x plus four.

2x²-3x+4=0

We find the roots by using quadratic equation formula.

x=-b±√b²-4ac/2a

a=2, b=-3, c=4

x=3±√9-4(8)/4

x=3±i√23/4

So the roots are x=3+i√23/4 and x=3-i√23/4.

A polynomial of degree 2 can have 2 real zeros, 1 real zero, or no real zeros

Hence, there are two zeros of the polynomial 2x²-3x+4.

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Eric rides his bike 4 1/3 hours during the week. His little brother rides his bike 1 3/4 hours during the week. How many more hours does Eric ride his bike during the week than his little brother? Enter your answer as a mixed number in simplest form by filling in the boxes.

Answers

The answer would be 2 7/12 (two and seven twelfths (seven over twelve))

What are 3 examples inequality?

Answers

Not fairness treated differently being separated from others by other’s opinions

What is value of expression example?

Answers

An example of value of expression is 5 × 2a.

In the field of mathematics, a value of an expression can be described as such an expression that includes two or more two mathematical numbers. Apart from the numbers, one or more than one mathematical operation is also a part of the value of the expression.

In the example, 5×2a, 5 and 2 are the two mathematical numbers and multiplication (×) is the mathematical operation. Hence, the expression 5×2a serves as an appropriate example of the value of expression in the field of mathematics.

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39 slices from 3 cakes = blank
slices per cake

Answers

Answer:

13

Step-by-step explanation:

39/3

If 3 cake has 39 slices, to find the slices that one cake has, we should take 39/3=13. Therefore one cake has 13 slices. Feel free to clarify if you’ve any questions!

Are Mean and median same?

Answers

Mean and Median are not the same.

Mean is the average value of a given dataset. Mean is calculated by adding all the values in the dataset and divided by the number of values.

Median is defined as the number in the middle of the dataset. Median is calculated by arranging the values in ascending order and finding out the middle most term. If there are two middle terms, then we calculate the average of both.

Mean is the arithmetic average, while median is the positional average. There may be various external factors that contribute in defining the mean. So for uneven data, median is the more robust value. Mean defines the central value of data, but median defines the midpoint of data and classify the data points to two sets, higher value or lower value.

So mean is the average of the given dataset and Median is the mid value of the dataset when arranged in ascending order.

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what is a polynomial function in standard form with zeros 1,2,-2 and -3

Answers

The standard form of the polynomial function with zeroes 1, 2, -2, and -3 is

[tex]x^4[/tex] + 2x³ - 7x² - 8x + 12 = 0.

What is a polynomial?

Polynomial is an equation written as the sum of terms of the form kx^n.

where k and n are positive integers.

We have,

The zeroes of a polynomial:

x = 1, 2, -2, and -3.

This can be written as,

(x - 1)(x - 2)(x + 2)(x + 3) = 0

Simplify to get the equation.

(x² - 2x - x + 2)(x² + 3x + 2x + 6) = 0

(x² - 3x + 2)(x² + 5x + 6) = 0

[tex]x^4[/tex] + 5x³ + 6x² - 3x³ - 15x² - 18x + 2x² + 10x + 12 = 0

[tex]x^4[/tex] + 2x³ - 7x² - 8x + 12 = 0

Thus,

The equation of the polynomial is [tex]x^4[/tex] + 2x³ - 7x² - 8x + 12 = 0.

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can anyone help me.

Answers

Answer:

Solution:

Given,

Ratio of photo to to the photo frame=2:3

Length of photo frame=36cm

= 36÷3

=13

then,

Length of photo = 13 ×2

= 26cm

Step-by-step explanation:

Therefore,the length of the photo is 26cm

Most graduate schools of business require applicants for admission to take the Graduate Management Admission Council’s GMAT examination. Scores on the GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112. What is the probability of an individual scoring above 500 on the GMAT?HOW HIGH MUST AN INDIVIDUAL SCORE ON THE G-MAT IN ORDER TO BE THE HIGHEST 5%

Answers

The probability of an individual scoring above 500 on the GMAT is 0.5984.

What is Normal Distribution?

An example of a continuous probability distribution is the normal distribution, in which the majority of data points cluster in the middle of the range while the remaining ones taper off symmetrically toward either extreme. The distribution's mean is another name for the centre of the range.

Given:

[tex]\mu[/tex] = 527

[tex]\sigma[/tex] = 112

Now, z= (500-527)/112

z= -27/112

z= -0.2410

and, P( X>500)= P(z> -0.2401)

                        = 1- 0.4052

                        = 0.5984

Hence, the probability of an individual scoring above 500 on the GMAT is 0.5984.

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The union of two sets is a set that contains only the elements that appear in both sets

a. True
b. False

Answers

False i am pretty sure

The union of two sets is to avoid counting the same elements twice.

What is set?

The mathematical logic subfield of set theory investigates sets, which can be loosely defined as collections of objects. Although any object can be combined into a set, set theory, as a mathematical discipline, focuses primarily on those that are relevant to mathematics as a whole.

Given union of set

(A ∪ B),

The set of all objects that are members of either A or B, or both, is the union of the sets A ∪ B.

let us take example,

A = { 1, 2, 3, 4}

B = {3, 4, 5, 6, 7}

(A ∪ B) = { 1, 2, 3, 4} ∪  {3, 4, 5, 6, 7}

we can simply write all of A and B's elements in a single set to avoid duplicates to find A U B.

(A ∪ B) = {1, 2, 3, 4, 5, 6, 7}

Hence the  union of two sets is a set that contains all the elements of both set and avoid duplicates.

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-X+4=67
-4Z-9=67
-7Y-10=67

Answers

Answer:

the answer is 20

Step-by-step explanation:

20+0=20

if quadrilateral abcd is rotated 90° clockwise about point a and then reflected across the y-axis to create quadrilateral a’b’c’d’, what will be the coordinate of d’?

Answers

According to the 90° clockwise rotation, the new coordinate of the quadrilateral D' is (3, -2)

Quadrilateral

Quadrilateral refers a closed shape and a type of polygon that has four sides, four vertices and four angles.

Given,

If quadrilateral ABCD is rotated 90° clockwise about point A and then reflected across the y-axis to create quadrilateral A’B’C’D’.

Here we need to find the coordinate of D'.

In order to find the new coordinates, first we have to identify the current coordinates of the quadrilateral.

While we looking into the graph, we have identify the point of

A (0,0), B(-4, 3), C(1,5) and D(2,3)

Now, we have to use the rule of clockwise 90° rotation, states that to rotate the figure 90 degrees clockwise about a point, every point(x , y) will rotate to (y, -x).

According to this rule, the new coordinates are written as,

A(0,0) = A'(0,0)

B'(-4,3) = B'(-3,-4)

C(1,5) = C'(5, -1)

D(2,3) = D' (3, -2)

Therefore, the new coordinate of D' is (3, -2).

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The parent function of the function g(x)=(x-h)² + kis f(x)=2. The vertex of the function g(x) is located at (9.-
8). What are the values of h and k?
g(x)=(x-____)^2+______

Answers

Answer:

h = 9

k = -8

Step-by-step explanation:

⭐ [tex]g(x) = (x-h)^2 + k[/tex] is already written in vertex form.

⭐ Vertex form: [tex]f(x) = a(x-h)^2 +k[/tex], where the vertex is (-(-h), k)

We are already given the vertex (9,-8), so we have to substitute these numbers into g(x).

[tex]g(x) = (x-9)^2 -8[/tex]

The equation of parabola is solved and the values of h and k are 9 and -8 respectively

What is a Parabola?

A Parabola, open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone. A parabola is a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line

The equation of the parabola is given by

( x - h )² = 4p ( y - k )

y = a ( x - h )² + k

where ( h , k ) is the vertex and ( h , k + p ) is the focus

y is the directrix and y = k – p

The equation of the parabola is also given by the equation

y = ax² + bx + c

where a , b , and c are the three coefficients and the parabola is uniquely identified

Given data ,

The vertex form of a parabola function is given by:

g(x) = a(x - h)² + k

where (h, k) is the vertex.

In this case, we are given that:

g(x) = (x - h)² + k

And the vertex is located at (9, -8)

So we have:

h = 9

k = -8

Therefore, the function g(x) can be written as:

g ( x ) = ( x - 9 )² - 8

Hence , the equation of parabola is g ( x ) = ( x - 9 )² - 8

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x and y are integers.
y>_2
x+y≤4
y≤ 2x + 2
Mark on the grid all the points
that fulfill all three inequalities.

Answers

The common region for all the inequalities y ≥ 2, y ≤ 2x + 2and x + y ≤ 4 is represented by ΔABC (the last graph).

What is an inequality?

"It is a statement which consists of an order relationship (greater than, greater than or equal to, less than, or less than or equal to) between two mathematical expressions."

For given question,

We have been given three inequalities

y ≥ 2 ,

x + y ≤ 4 ,

y ≤ 2x + 2

If we draw the graph of these three inequalities then the common region represents the all the points that fulfill all three inequalities.

An inequality y ≥ 2 represents all the points on and above the line y = 2

An inequality x + y ≤ 4 represents all the points on and below the line       x + y = 4

An inequality y ≤ 2x + 2 represents all the points on and below the line    y = 2x + 2

Consider the following graphs.

The first graph represents the points which fulfill the inequality y ≥ 2

The graph for an inequality x + y ≤ 4 is represented by the shaded blue region.

The graph for an inequality y ≤ 2x + 2 is represented by the shaded green region.

The common region for all the inequalities is represented by ΔABC

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Is 9 by 16 equal to 5 by 9 yes or no?

Answers

Answer:

No

Step-by-step explanation:

9/16 and 5/9 have a common factor is 144

So, we have

81/144  and 80/144

So, 9/16 > 5/9

Answer:

Step-by-step explanation:

no they are not equal, NO

Solve the system of equations.
−5y+3x=3
−8y+9x=−12

Answers

Answer:

The solution to the system of equations is x = 9/12 and y = 27/60.

Step-by-step explanation:

To solve a system of equations, we can use elimination or substitution. In this case, we can use elimination by adding the equations together. When we add the two equations, the y-terms will cancel out, leaving us with an equation in terms of x only.

First, let's add the equations:

-5y + 3x = 3

-8y + 9x = -12

-----------

-13y + 12x = -9

Then, we can divide both sides of the equation by -13 to get the value of x:

(-13y + 12x) / -13 = (-9) / -13

12x = 9

x = 9/12

Since we know the value of x, we can substitute it back into either of the original equations to find the value of y:

-5y + 3(9/12) = 3

-5y + 27/12 = 3

-5y = -27/12

y = 27/60

Therefore, the solution to the system of equations is x = 9/12 and y = 27/60.

What is mode 2 2 3 3 4 4 5 5 6 6 is?

Answers

The given data (2, 2, 3, 3, 4, 4, 5, 5, 6, 6) has no mode.

A mode is defined as the value that has the highest frequency in a given set of values. It is the value that appears the most number of times.

In statistics, the value that consistently appears in a particular collection is referred to as the mode. The mode or modal value in a data collection is sometimes referred to as the value or number that occurs most frequently in the data set. In addition, to mean and median, it is one of the three measures of central tendency.

The mode is the value that occurs the most often in a given set of data.

Given data:  2, 2, 3, 3, 4, 4, 5, 5, 6, 6.

In the given observations, data is repeated an equal number of times.  

So, the given data has no mode.

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Why we need to clean the measuring instrument before and after use?

Answers

Conscientiously cleaning your devices is crucial, as any residue that remains on your device may impact the accuracy of your measurements. The material your device is made of plays a significant role in the cleaning method you should use.

A tool used to measure a physical amount is called a measuring instrument. Measurement is the process of gathering and comparing physical quantities of actual things and events in the physical sciences, quality control, and engineering.

The process of measuring yields a number associating the subject under study and the referenced unit of measurement, and established standard objects and occurrences are employed as units. These numerical relations are acquired using measuring devices and rigorous test procedures that specify the instrument's use.

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How many days does SSS approved disbursement account?

Answers

According to the SSS disbursement account, it takes 2 business days for approval.

The term disbursement refers to the payment of federal student aid funds to the student by the school.

Here we have to find the number of days taken for SSS disbursement account approval.

In order to find the solution, we must know the process of SSS disbursement policy, that takes the payment data and transforms it into a disbursement instrument.

Here we have given that, as per the SSS disbursement policy, it takes member-borrower shall receive the full short-term loan proceeds in their account within two banking days from the loan date provided the payment instruction of SSS was sent before the cut-off time.

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What is a possible factor of x³ 6x² 11x 6?

Answers

Therefore , the solution to the given problem of the equation whose factors are (x+2)(x+1)(x+3).

Describe equation.

Two expressions joined by an equal sign form a mathematical statement known as an equation.

An equation is, for instance, 3x - 5 = 16. We can solve this equation and determine that the value of the variable x is 7.

According to the Given question:

x³ + 6x² + 11x +6

This pattern of coefficients 1, 6, 11, 6 is actually one that you are likely to encounter frequently.

For demonstration reasons, let's see what occurs if we begin by using the typical technique for resolving a general cubic:

We can use a linear substitution, t = x + b/3a, to simplify any cubic of the form ax3 + bx2 + cx + d, resulting in a cubic in the form a + b + et + f without a quadratic term.

This is the most straightforward type of Tschirnhaus transformation.

In our example, a = 1 and b = 6, so we want t = x + 2.

Note that:

=  + 6+ 12x +8

So we find:

+ 6+ 11x +6 =   − (x + 2)

=  - t

= t ( - 1)

= t(t − 1)(t + 1)

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

Therefore , the solution to the given problem of the equation whose factors are (x+2)(x+1)(x+3).

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A new bank customer with ​$4,500 wants to open a money market account. The bank is offering a simple interest rate of 1.5​%.

Answers

Answer:

a. 1350

b. 6500

Step-by-step explanation:

1.5%×4500=5850

1350+4500=5850

Step-by-step explanation:

What does a standard deviation of 35 mean?

Answers

As the standard deviation 35 is high so the values are far from the mean.

What is standard deviation?

In the statistics, standard deviation is an estimation that implies how far each available data apart from mean value. In a word, it is a measure of variances of data in relation to the mean. A low standard deviation indicates that values are closer to mean value and high standard deviation indicates the values are distributed widely.

What does a standard deviation of 35 mean?

Standard deviation of a given data set 35 means that all data set are 35 unit far from the mean value. As the value is high so all the values are dispersed widely relative to the mean value.

In the other hand, this high standard deviation implies there are enough variances in the given data in relation to the mean.

Standard deviation is one of the measures of risk assessment. Imagine an organization published a number of statistical observation data. If we find a high standard deviation, then it will be high risk of collapse of the business.

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How do you copy a formula in Excel?

Answers

To see the steps of copy and paste a formula.

Now, According to the question:

Here's how you copy and paste a formula:

1. Select the cell with the formula you want to copy.

2. Press. + C.

3. Click the cell where you want to paste the formula.

4. To quickly paste the formula with its formatting, press + V.

5. Clicking the arrow gives you a list of options.

Let's know to more:

How do I apply the same formula to multiple cells in Excel?

Fill formulas into adjacent cells

1. Select the cell with the formula and the adjacent cells you want to fill.

2. Click Home > Fill, and choose either Down, Right, Up, or Left. Keyboard shortcut: You can also press Ctrl+D to fill the formula down in a column, or Ctrl+R to fill the formula to the right in a row.

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Which formula can be used to find the nth term of the geometric sequence 1/6,1,6,36?

Answers

Answer:

[tex]a_n = \frac{1}{6}(6)^{n-1}[/tex]

Step-by-step explanation:

What is the formula of a geometric sequence/progression?

[tex]a_n = a_1+1(d)^{n-1}[/tex][tex]a_1[/tex] is the first term of the geometric sequence[tex]d[/tex] is the common ratio (the number each consecutive term gets multiplied by) of the geometric sequence[tex]n[/tex] is the number term you want to find

We are given the geometric sequence.

Thus, we need to find the first term and the common ratio, and substitute those values into the geometric sequence formula.

The first term is [tex]\frac{1}{6}[/tex]

How do we find the common ratio?

divide one term by another termmake sure both terms are consecutive of each other

I am choosing term 1 and term 2, but you can choose any other 2 terms that are consecutive.

term 1 = [tex]\frac{1}{6}[/tex]

term 2 = [tex]1[/tex]

[tex]\frac{term2}{term1} = \frac{1}{\frac{1}{6}}[/tex]

[tex]\frac{term2}{term1} = 6[/tex]

∴ The common ratio of this sequence is 6.

Now, substitute these values (common ratio & first term) into the geometric sequence equation.

[tex]a_n = \frac{1}{6}(6)^{n-1}[/tex]

What is the y-intercept of 5x 4y =- 12?

Answers

The y-intercept of the equation 5x-4y = -12 is 3.

What is a slope?

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

We have the equation:

5x-4y = -12

To find the y-intercept:

First, we have to simplify the equation as subject to y.

5x-4y = -12

4y = 5x+ 12

y = (5/4)x+12/4

y =  (5/4)x + 3

Now, we know the slope-intercept form:

y = mx+b,

where m is the slope and b is the y-intercept.

Comparing, both the equation,

b = 3

3 is the y-intercept.

Therefore, y-intercept is 3.

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Renting a roller rink for your birthday costs $250 plus $10 per person. The total charge
for your party was $400. Translate this information into an equation using a to
represent the number of guests at your party.
Using your equation, determine how many people attended your party.

Answers

Answer:

15 people attended the party

the equation would be 250+10x=400
250 for renting the space and 10 per person and x represents the amount of people and 400 was the total with the rent and the cost for the total amount of people

step by step of solving that would be
250+10x=400
-250 -250
10x=150
10/10 150/10
x=15

so there were 15 people at the party

Which exponential function has a growth factor of 5?

Answers

So, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

An exponential function of the form y = a * b^x has a growth factor of b. Therefore, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

For example, if a = 2, then the function y = 2 * 5^x has a growth factor of 5.

Here is a table showing the values of y for some values of x:

x       y

0        2

1        10

2        50

3        250

4        1250

As x increases, the values of y increase exponentially at a rate determined by the growth factor of 5.

Therefore, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

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For example, if a = 2, then the function y = 2 * 5^x has a growth factor of 5.

Here is a table showing the values of y for some values of x:

x       y

0        2

1        10

2        50

3        250

4        1250

As x increases, the values of y increase exponentially at a rate determined by the growth factor of 5.

Therefore, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

Can yall please help me with this I really need help

Answers

Answer:

i believe it is G

[tex] \frac{7}{3} [/tex]

A baketball player attemptd 15 baket in a game he make 9 which ratio decribe the number of baket the player made to the number of baket the playerd attempted

Answers

The ratio that describes the number of baskets the player made to the number of baskets the player attempted is 9:15. This can be simplified to 3:5.

The ratio of the number of baskets made by the player to the number of baskets attempted is 9/15. This ratio can be simplified to 3/5, which means that for every 5 baskets attempted, the player made 3 baskets. This ratio can also be expressed as a percentage by dividing the number of baskets made by the number of baskets attempted and multiplying the result by 100%. In this case, the percentage of baskets made by the player is 60% (9 baskets made / 15 baskets attempted * 100% = 60%). This means that out of every 100 baskets attempted by the player, he made 60 baskets.

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Other Questions
A number split by 3 in algebraic expression What happens in chapter 3 of Huckleberry Finn? What are the 3 types of sprint? What process produces minerals in the oyster shell in photo B? Where do the ions for these minerals come from? Choose one: A. The ions are gained during compaction of the sediment. B. The ions come from the soft parts of the animal. C. The animal is able to extract the ions from the water it lives in. D. The ions come from the food the animal eats. A man is four times as old as his son. In 5 years time he will be three times as old as his son. What is the present age of the son in years? What agile really means? What does infinity x mean? What is public property give example? Which one is a better measure of the economic performance of a country? An equation is given.y=3x+cCreate an equivalent equation by solving for c in terms of y and x. What values are important in Vietnamese culture? An author is writing an educational explanation of a historical event for a textbook. What type of syntax would best fit their purpose? aLong sentences with plenty of commas to reflect lengthy, emotional thoughts bMedium sentences with minimal punctuation to reflect a straightforward explanation cShort sentences with dashes and exclamation marks to reflect short, nervous bursts dSimple sentences that lack details and are difficult for the reader to follow What was the main reason for the Latin American revolution? R r, ?R=3;r=0,5;3. a researcher wants to study students to see if age of students has a positive relationship with grade point average. the researcher will get both age and grade point average from the college but not identifying individual students. grade point average appears to follow the normal curve. which test statistic would you use? what is a predicate function? group of answer choices a function that returns a boolean result. a function that has special meaning in a class. a function that takes no arguments. a function that calls another function. What key elements needs to be addressed in Jupiter Bachs corporate culture for the new sustainability strategy to have an impact on the performance When Marco was 18 years old, his friend, Yohan was 9. How old will Yohan be when Marco is 24? Given the following two ordered pairs calculate the rate of the function.(9, 4) and (3, 16)Enter your answer in the box.m= What piece of information from Passage 1 conflicts with the information from Passage 2? A. People should only eat fruits and vegetables as it is the healthiest food and nutrient-dense food available. B. People normally have a difficult time digesting food, so the only solution is to eat fruits and vegetables. C. Eating as many fruits and vegetables as part of ones diet is a very healthy way of maintaining ones well-being. D. Eating fruits and vegetables is like medicine and will cure most chronic diseases and reduce blood cholesterol.