Consider a linear transformation T from R^3 to R^2 for which T = [\begin{array}{ccc}1\\0\\0\end{array}\right] = [\begin{array}{ccc}1\\9\end{array}\right],
T = [\begin{array}{ccc}0\\1\\0\end{array}\right] = [\begin{array}{ccc}5\\8\end{array}\right],,
T = [\begin{array}{ccc}0\\0\\1\end{array}\right] = [\begin{array}{ccc}6\\0\end{array}\right],
Find the matrix A of T. A=[____]
Answers
A linear transformation T from R^3 to R^2 can be represented by a 2x3 matrix A. Given the information,
\(T([1, 0, 0]^T) = [1, 9]^T and T([0, 0, 1]^T) = [6, 0]^T.\)
To find the matrix A, we can express these transformations in terms of the standard basis vectors of R^3, which are
\(e1 = [1, 0, 0]^T, e2 = [0, 1, 0]^T\), and e3 = [0, 0, 1]^T.
The transformation T is applied to these basis vectors, and the resulting vectors form the columns of the matrix A.
We already have the information for T(e1) and T(e3):
\(T(e1) = [1, 9]^T and T(e3) = [6, 0]^T.\)
However, we don't have the information for T(e2). In general, we can write the matrix A as:
\(A = [[a, b, c], [d, e, f]],\)
where \(T(e1) = [a, d]^T, T(e2) = [b, e]^T,\) and \(T(e3) = [c, f]^T.\)
Using the given transformations, we have:
\(T(e1) = [1, 9]^T\), so a = 1 and d = 9.
\(T(e3) = [6, 0]^T\), so c = 6 and f = 0.
Unfortunately, we cannot determine the exact values for b and e, since we don't have information about T(e2). However, we can partially fill in the matrix A:
\(A = [[1, b, 6], [9, e, 0]].\)
Without information about T(e2), we cannot complete the matrix A.
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Related Questions
Which equations have a lower unit rate than the rate represented in this table?
х у
6 2
12 4
18 6
y=3/11x
y=2/5x
y=9/23x
y=4/13x
y=3/8x
Answers
Answer:
A and DStep-by-step explanation:
Find the unit rate of the data in the table:
y = kx2 = 6kk = 2/6k = 1/3The equation is:
y = 1/3xCompare this with the others:
A. 3/11 < 1/3 = 3/9 ⇒ yesB. 2/5 > 1/3 = 2/6 ⇒ noC. 9/23 > 1/3 = 9/27 ⇒ noD. 4/13 < 1/3 = 4/12 ⇒ yesE. 3/8 > 1/3 = 3/9 ⇒ noFind. rate
m=4-2/12-6m=2/6m=1/3Only first one and 4the one have lower unit rate
y=3/11xy=4/13xWhat is the exact height of a right triangle with an angle that measures 30 degrees adjacent to a base of 4.
Answers
Answer:
2.3
Step-by-step explanation:
We are to determine the opposite length given an angle of 30 degrees and an adjacent side of 4
we would make use of tan 30
tan 30 = opposite / adjacent
0.5774 = opposite / 4
opposite = 0.5774 x 4 = 2.3
two cables are connected to the top of a very tall pole and are pulled tight in opposite directions, then connected the the ground. one cable is 48 feet long, and the other is 63 feet long. the ground distance between them is 80 feet. how tall is the pole, measured to the nearest tenth?
Answers
The height of the pole is approximately 64 feet when rounded to the nearest tenth.
To determine the height of the pole, we can use the concept of a right triangle formed by the pole and the two cables. Let's denote the height of the pole as 'h'.
In the given scenario, one cable is 48 feet long and the other is 63 feet long. The ground distance between them is 80 feet. We can visualize this as follows:
A
/|
/ |
h / | 63
/ |
/ |
/ |
/______C
48 B
Here, A represents the top of the pole, B represents the point where the 48-foot cable touches the ground, and C represents the point where the 63-foot cable touches the ground.
Using the Pythagorean theorem, we can establish the following relationship:
\(AB^2 + BC^2 = AC^2\)
Substituting the given values, we get:
\(h^2 + 48^2 = 80^2\\h^2 + 2304 = 6400\\h^2 = 6400 - 2304\\h^2 = 4096\)
Taking the square root of both sides, we find:
h =\(\sqrt{4096}\)
h ≈ 64
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Suppose that the water level of a river is 34 feet and that it is receding at a rate of 0.5 foot per day. Write an equation for the water level, L, after d days.
Answers
Answer:
L = -0.5x + 34
Step-by-step explanation:
When finding m in the formula y = mx + b. Find a word that means multiplication, in this case 0.5 feet per day.
a number that can be expressed as a ratio of two integers is called ____?
Answers
A number that can be expressed as a ratio of two integers is called rational number.
There are four type of numbers, namely natural, whole, rational numbers and integers. Natural numbers begin from 1 and end at infinity. The whole numbers are 0 and all the natural numbers.
Integers are of further two types, positive and negative numbers. The numbers represented on number line are integers. Rational numbers are the numbers that can be represented as p/q or as fraction, where denominator should not be zero.
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The joint probability density function of x and y is given by f(x,y)=(x+y)/8 0 calculate the variance of (x+y)/2
Answers
The variance of (X + Y)/2 is 5/9. The variance of a random variable is defined as E[(X - E[X])^2], where E[X] is the expected value of the random variable.
We are given the joint probability density function of X and Y as f(x,y) = (x+y)/8 for 0 < x < 2 and 0 < y < 2.
We need to find the variance of (X + Y)/2.
Let Z = (X + Y)/2. Then, the expected value of Z is:
E[Z] = E[(X + Y)/2] = E[X]/2 + E[Y]/2
We can find E[X] and E[Y] as follows:
E[X] = ∫∫x f(x,y) dxdy = ∫∫x(x+y)/8 dxdy
= ∫0²∫0²x(x+y)/8 dydx = 2/3
Similarly, E[Y] = 2/3.
Therefore, E[Z] = 2/3.
Now, we need to find E[Z^2]:
E[Z^2] = E[(X + Y)^2/4] = E[X^2]/4 + E[Y^2]/4 + E[XY]/2
We can find E[X^2], E[Y^2], and E[XY] as follows:
E[X^2] = ∫∫x^2 f(x,y) dxdy = ∫0²∫0²x^2(x+y)/8 dydx = 2
E[Y^2] = ∫∫y^2 f(x,y) dxdy = ∫0²∫0²y^2(x+y)/8 dydx = 2
E[XY] = ∫∫xy f(x,y) dxdy = ∫0²∫0²xy(x+y)/8 dydx = 1/2
Therefore, E[Z^2] = (2/4) + (2/4) + (1/4) = 1.
Finally, we can find the variance of Z as:
Var(Z) = E[Z^2] - E[Z]^2 = 1 - (2/3)^2 = 5/9.
Therefore, the variance of (X + Y)/2 is 5/9.
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What is the result when the number 31 is increased by 8% round your answer to the nearest 10th
Answers
The result when the number 31 is increased by 8% and rounded to the nearest tenth is 33.5.
To find the result when the number 31 is increased by 8%, we can calculate 8% of 31 and add it to 31.
8% of 31 can be found by multiplying 31 by 0.08:
8% of 31 = 31 * 0.08 = 2.48
Now, we add this result to 31:
31 + 2.48 = 33.48
Rounding this answer to the nearest tenth, we get:
33.5
Therefore, the result when the number 31 is increased by 8% and rounded to the nearest tenth is 33.5.
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Consider the set N2 N x N, the set of all ordered pairs (a, b) where a and b are natural numbers. Consider a function f: N2 N given by f((a, b)) a b {(a, b) E N a, b < 10. Find f(A) a. Let A b. Find f1(3) and f1({0,1,2,3}) c. Give geometric descriptions of f1(n) and f1({0,1,... , n}) for any n 2 1. d. Find |f(8) and If1(0,1, ,8})|
Answers
a. f1(3) = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. b. f1({0, 1, 2, 3}) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.c. Geometric descriptions a set of horizontal lines in the xy-plane. d. |f(8)| = 19 and |f1({0, 1, ..., 8})| = 13.
To find f(A) where A = {(a, b) | a, b ∈ N, a, b < 10}, we need to apply the function f to each element in A.
f((a, b)) = a + b
So, let's evaluate f for each element in A:
f((0, 0)) = 0 + 0 = 0
f((0, 1)) = 0 + 1 = 1
f((0, 2)) = 0 + 2 = 2
f((9, 7)) = 9 + 7 = 16
f((9, 8)) = 9 + 8 = 17
f((9, 9)) = 9 + 9 = 18
Therefore, f(A) = {0, 1, 2, ..., 16, 17, 18}.
a. To find f1(3), we need to apply the function f to the ordered pair (3, b) for b = 0, 1, 2, ..., 9.
f1(3) = {f((3, 0)), f((3, 1)), f((3, 2)), ..., f((3, 9))}
= {3 + 0, 3 + 1, 3 + 2, ..., 3 + 9}
= {3, 4, 5, ..., 12}
Therefore, f1(3) = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
b. To find f1({0, 1, 2, 3}), we need to apply the function f to the ordered pairs (0, b), (1, b), (2, b), and (3, b) for b = 0, 1, 2, ..., 9.
f1({0, 1, 2, 3}) = {f((0, 0)), f((0, 1)), f((0, 2)), ..., f((3, 9))}
= {0 + 0, 0 + 1, 0 + 2, ..., 3 + 9}
= {0, 1, 2, ..., 12}
Therefore, f1({0, 1, 2, 3}) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
c. Geometric descriptions of f1(n) and f1({0, 1, ..., n}) for any n ≥ 1:
- f1(n): This represents a set of horizontal lines in the xy-plane. Each line is defined by a constant y-value, ranging from 0 to n. The lines are parallel to the x-axis and are equally spaced with a distance of 1 between each line. The intersection points of these lines with the x-axis correspond to the values in f1(n).
- f1({0, 1, ..., n}): This represents the filled region between the x-axis and the lines described in f1(n). It forms a trapezoidal shape in the xy-plane, where the base of the trapezoid is the x-axis and the top side of the trapezoid is formed by the lines defined in f1(n). The vertices of this trapezoid are located at (0, 0), (n, 0), (n,
n), and (0, n), with the lines defined in f1(n) forming the top side of the trapezoid.
d. To find |f(8) and |f1({0, 1, ..., 8})|, we need to determine the cardinality (number of elements) of the respective sets.
|f(8)| = 19 (since f(8) = {0, 1, 2, ..., 16, 17, 18} and it contains 19 elements).
|f1({0, 1, ..., 8})| = 13 (since f1({0, 1, ..., 8}) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and it contains 13 elements).
Therefore, |f(8)| = 19 and |f1({0, 1, ..., 8})| = 13.
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How are the properties of exponents used when dividing a polynomial by a monomial?
Answers
Answer:
a couple different obes
Step-by-step explanation:
there are five
how do we decide whether to use a t test or an analysis of variance (anova) to analyze the data from an experiment?
Answers
The best way to analyze their data for significance is to Conduct a two-way variable analysis.
As we can see, there is a 2x3 factorial design. Thus, we know that there will be a two level or two way analysis of variables, which is referred to as two way variables analysis. Factorial design is a research method that allows for the inquiry of a main and interaction effects of two or more independent variables on one or more outcome variables (s).
It has been argued that factorial designs represent the true beginning of modern behavioral research and have resulted in a significant paradigm shift in how social scientists conceptualize their research questions and produce objective results (Kerlinger & Lee, 2000).
Factorial design is an experimental methodology that goes beyond standard single-variable experimentation. Previously, social scientists were fixated on single independent variable experiments, foreshadowing the significance of extraneous variables that can attenuate or diminish research findings.
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Write an equation that represent the line
Answers
Which measure of central tendency is most appropriate when it is desired to minimize the effect of a few extreme (very high or very low) scores?group of answer choicesmedian
Answers
When it is required to minimize the effect of a few extreme scores, median is the measure of central tendency which is the most appropriate.
This is due to the fact that the median isn't influenced by extreme values, which means it is the middle value when the observations are ordered from least to greatest.There are several measures of central tendency, including the mode, median, and mean.
The most common measure is the mean, but it is greatly affected by the presence of a few very high or very low scores.To ensure that the mean isn't skewed too much by these scores, the median is a better choice. The median is the middle number in a data set when the observations are sorted from smallest to greatest.
In this way, if there are extremely high or low scores, they are not going to influence the median significantly as compared to the mean. Hence, we can say that the median is the most appropriate measure of central tendency when it is desired to minimize the effect of a few extreme scores.
The measure of central tendency that is most appropriate when it is desired to minimize the effect of a few extreme (very high or very low) scores is the median.
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Which of the following is the surface area of the right cylinder below?
A. 204units2
B. 480 units2
C. 408 units2
D. 240ft units2
SIIRMIT
Answers
Answer:
There is no photo all I can say is to re ask this question so that we are able to see a photo.
Step-by-step explanation:
Match each multiplication problem with the
answer.
Answers
Answer:
1. D
2.C
3. A
4. B
Step-by-step explanation:
times each of the numbers by however many r in the brackets
3×2=6
3×-1=-3
so the answer to 1 will be (6)
(-3)
A line has a slope of negative 2/3 and passes through the point (-3, 8 ) what is the equation of the line
Y= - 2/3x+ six
Y= - 2/3x+8
Y= 6x- 2/3
Y= 8x- 2/3
Answers
Answer:
Y= -⅔x +6
(1st option)
Step-by-step explanation:
y=mx +c
m is the gradient
c is the y-intercept
Given that the slope is -⅔, m= -⅔.
Substitute m= -⅔ into the equation:
y= -⅔x +c
To find the value of c, substitute a pair of coordinates.
When x= -3, y= 8,
\(8 = - \frac{2}{3} ( - 3) + c \\ 8 = 2 + c \\ c = 8 - 2 \\ c = 6\)
Thus, the equation of the line is y= -⅔ +6.
So you are given point (3, -4) and slope 2. y=mx+b is the basic equation for linear graphs, and in this equation, b is the y-intercept, m is the slope, and y and x are where you can substitute your points.
What is 3x+y over z when x=21, y = 36, and z= 49?
Answers
Answer:
Step-by-step explanation:
2.02
x=21 so replace x with 21
3(21)+y/z
then do the same with y and z
3(21)+36/49
63+36/49
99/49
SOLUTION 2.02
Tonya's income is four times as much as Nora's income. Write an Algebraic expression representing Nora's income in terms of Tonya's
Answers
An algebraic expression for representing the Nora's income in form of Tonya's income is given by y = ( x / 4 ) .
Let us consider 'x' represents the Tonya's income.
And variable 'y' represents the Nora's income.
Tonya income is equal to four times of Nora's income.
This implies,
Nora's income is equal to one fourth times of Tonya's income.
⇒ y = ( x / 4 )
Rewrite an algebraic expression to represents Tonya's income in terms of Nora's income we have,
Simplify by multiplying both the sides of the algebraic expression by 4 we get,
⇒ x = 4y
Therefore, an algebraic expression to represents the Nora's income in terms of Tonya's income is equal to y = ( x / 4 ).
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There is a bag with only red marbles and blue marbles.
The probability of randomly choosing a red marble is 2/7
There are 16 red marbles in the bag and each is equally likely to be chosen.
Work out how many marbles in total there must be.
Answers
Answer: 56
Step-by-step explanation:
Given that :
Marbles in bag are only red and blue ;
Probability of randomly choosing a red = 2/7
Number of red marbles in bag = 16
The total Number of marbles =?
P(choosing a red marble) = required outcome / Total possible outcomes
Required outcome = number of red marbles
Total possible outcomes = total number of blue and red marbles
Total number of blue and red marbles = x
P(choosing a red marble) = required outcome / Total possible outcomes
2/7 = 16 / x
Cross multiply
2x = 16 * 7
2x = 112
x = 112 / 2
x = 56 marbles
Total number of marbles = 56
A cyclist rides her bike at a rate of 12 kilometers per hour. what is the rate in kilometers per minute? how many kilometers will the cyclist travel in 20 minutes? do not round your answers.
Answers
The cyclist will travel 4 kilometers in 20 minutes.
To find the rate in kilometers per minute, we can divide the rate in kilometers per hour by 60, since there are 60 minutes in an hour. Therefore, the rate in kilometers per minute is:
12 km/h ÷ 60 min/h = 0.2 km/min
To find how many kilometers the cyclist will travel in 20 minutes, we can use the formula:
distance = rate x time
where rate is in kilometers per minute, and time is in minutes. Plugging in the values, we get:
distance = 0.2 km/min x 20 min = 4 km
Therefore, the cyclist will travel 4 kilometers in 20 minutes.
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Could these triangles be congruent?
B
E
O yes,
O yes, if AB - DE
s, if BC = 7
O yes, if AB EF
O no, because the hypotenuses must have different
lengths
A
7
С
D
7
F
Answers
Answer:
yes, if AB ≅ DE
Step-by-step explanation:
Triangles are said to be congruent if they have the same sides and the same angles.
The following measures are used to determine if triangles are congruent:
1) Angle-side-angle: If two angles and a side of a triangle is equal to two angles and corresponding side of another triangle, then they are congruent.
2) Side-side-side: If all three sides of a triangle is equal to three sides of another triangle, then the two triangles are congruent.
3) Side angle side: If two sides and an included angle of a triangle is equal to the two sides and corresponding angle of another triangle, then they are congruent.
4) Hypotenuse - leg: If the hypotenuse and one leg of a triangle is equal to the hypotenuse and leg of another triangle then they are congruent.
From the triangles DEF and ABC, already, they already have one equal triangle that is ∠F = ∠B and an equal side i.e DF = AC.
To satisfy congruence, two sides and an angle have to be equal, therefore if AB = DE then the two triangles would be congruent
Answer:
D). no, because the hypotenuses must have different lengths
Step-by-step explanation:
I just did the assignment on Edge and it's 100% correct.
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Can you help me with this please. It was question on my test but it didn't make any sense since i got a repeating decimal.
3.5g-2.7g=-6
Answers
The solution of the linear equation 3.5g - 2.7g = -6 will be negative 7.5.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
If the maximum power of the equation is one. Then the equation will be a linear equation.
The linear equation is given below.
3.5g - 2.7g = -6
Simplify the equation, then we have
3.5g - 2.7g = -6
0.8g = -6
g = - 6 / 0.8
g = - 7.5
The solution of the linear equation 3.5g - 2.7g = -6 will be negative 7.5.
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In a certain library the first shelf is 15cm off the ground, and the remaining four shelves are each spaced 38.0cm above the previous one. If the average book has a mass of 1.40kg with a height of 22.0cm, and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves assuming the books are all laying flat on the floor to start?
Answers
The total work required to fill all the shelves by multiplying the work done for each shelf by the total number of shelves: Total work = Work per shelf x Number of shelves= 61.6 J x 4 shelves= 246.4 J Therefore, it requires 246.4 joules of work to fill all the shelves.
The library shelf is made up of five shelves. The first shelf is 15 cm off the ground, and the other four shelves are spaced 38 cm apart. The average book has a height of 22 cm and a mass of 1.40 kg.
Each shelf can hold up to 28 books placed vertically, To solve this problem, we need to find the total amount of work required to move all books from the floor to the shelves. We can calculate the work done using the following formula:
Work = Force x Distance Since the books are initially lying flat on the floor, we need to lift them up to each shelf.
Therefore, the force required to lift the books is equal to the weight of all the books, which is given by:Force = mass x gravity where gravity is the acceleration due to gravity and is equal to 9.81 m/s^2.
The mass of each book is 1.40 kg, and the total number of books that can be stored on each shelf is 28 books.
Therefore, the mass of books on each shelf is given by:
mass of books on each shelf = 1.40 kg/book x 28 books= 39.2 kgNow, we can calculate the force required to lift the books on each shelf using:Force = mass x gravity= 39.2 kg x 9.81 m/s^2= 385 N
To find the work required to lift all the books onto each shelf, we need to multiply the force required by the distance lifted.
The distance lifted for each shelf is given by:Distance = shelf height - book height= 38 cm - 22 cm= 16 cm= 0.16 m
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Erika went to the store. She got a pencil for 16¢ and a tablet for 25¢. She gave the storekeeper 50¢. How much more money did she get back?
Answers
Answer:
9c
Step-by-step explanation:
16+25=41
50-41=9
fnfnx
What is an equation of the line that passes through the points (-3, 8) and (-2, 3)?
Answers
Answer:
y = -5x - 7
Step-by-step explanation:
(Change in y) / (Change in x)
(3-8) / (-2 - (-3))
-5 / 1
-5 is the slope
So far we have y = -5x
Plug in a set of values and find the difference
3 = -5(-2)
3 = 10
This is false
We need to subtract 7
Our equation therefore is
y = -5x - 7
Momo has $4.05 in dimes and quarters. If she has 5 more quarters than dimes,
how many of each does she have? Show your work and write a complete
sentence answer
Answers
Answers:
8 dimes13 quarters==================================================
Work Shown:
$4.05 = 405 cents
d = number of dimes
q = number of quarters
q = d+5 since she has 5 more quarters compared to dimes
10d = value in cents of all the dimes
25q = value in cents of all the quarters
10d+25q = total value in cents = 405
10d+25q = 405
10d+25(d+5) = 405
10d+25d+125 = 405
35d+125 = 405
35d = 405-125
35d = 280
d = 280/35
d = 8
She has 8 dimes
q = d+5 = 8+5 = 13
She also has 13 quarters
-------------------------
1 dime = 10 cents
8 dimes = 80 cents (multiply both sides by 80)
1 quarter = 25 cents
13 quarters = 325 cents (multiply both sides by 13)
8 dimes + 13 quarters = 80 cents + 325 cents = 405 cents
This converts back to $4.05, which confirms our answers.
THE PENGUIN PATTERN Can you figure out the value of the '2' in the pattern below? ( 1 3 7 13 21 ? 43
Answers
Answer: The answer is 31
Step-by-step explanation:
Why because if you look closely at the starting of the pattern it starts with one then adds 2 for every number it gets. so when I reached 21 it added 10 to make 31 then added 12 to get 43 I hope you understand.
prove that if two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. (theorem)
Answers
Two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
Given,
Sides of quadrilateral are parallel and congruent .
Now,
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is said to be parallelogram.
Mathematically,
If Line AB ≅ Line CD and Line BC ≅ Line DA , then ABCD is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is said to be parallelogram.
Thus by the use of congruent properties we can state that quadrilateral having opposite sides parallel and congruent is a parallelogram.
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Which systems of equations have no solution?
Answers
Answer:
The last block where is says y=13-2x and 4x-y=-1 is the correct answer
I got a 100% in this test. It is right
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For the expression below, combine like terms to simplify the expression. -4x + 3y - 6 -15x + 23y - 10
Answers
Answer: -19x + 26y - 16
Step-by-step explanation:
Combine like terms:
= -4x + 3y - 6 -15x + 23y - 10
= -4x - 15x + 3y + 23y - 6 - 10
Simplifying the expression:
= -4x - 15x + 3y + 23y - 6 - 10
= -19x + 26y - 16
Find the general solution of the given differential equation.
(x + 1) dy/dx + (x + 2)y = 2xe^-x
y=
Answers
The solution involves an integral that cannot be evaluated in closed form, so the answer cannot be simplified further.
How to solve the given differential equation (DE)?To solve the given differential equation (DE), we can use the integrating factor method. The steps are as follows:
1. Multiply both sides of the DE by the integrating factor, which is the exponential of the integral of the coefficient of y. In this case, the coefficient of y is (x + 2), so the integrating factor is e^(∫(x+2)dx) = e^(x^2/2 + 2x).
So, we have: (x + 1) e^(x^2/2 + 2x) dy/dx + (x + 2) e^(x^2/2 + 2x) y = 2x e^(x^2/2 + 2x) e^(-xy)
2. Notice that the left-hand side of the DE is the product of the derivative of y with respect to x and the integrating factor, so we can apply the product rule of differentiation to obtain:
d/dx [ e^(x^2/2 + 2x) y ] = 2x e^(x^2/2 + 2x) e^(-xy)
3. Integrate both sides of the previous equation with respect to x to obtain:
e^(x^2/2 + 2x) y = - e^(-xy) + C
where C is the constant of integration.
4. Solve for y by dividing both sides by the integrating factor:
y = [- e^(-xy) + C] e^(-x^2/2 - 2x)
This is the general solution of the given DE.
Note that the solution involves an integral that cannot be evaluated in closed form, so the answer cannot be simplified further.
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