let r be a ring and r1,...,rn ∈ r. prove that the subset ⟨r1,...,rn⟩={λ1r1 ··· λnrn |λ1,...,λn ∈ r}isanidealin r.
Answers
By the above closure under subtraction and commutativity with ring elements, the subset ⟨r1,...,rn⟩={λ1r1 ··· λnrn |λ1,...,λn ∈ r} is an ideal in r.
Given that r be a ring and r1, ..., rn ∈ r. We need to prove that the subset ⟨r1,...,rn⟩={λ1r1 ··· λnrn |λ1,...,λn ∈ r} is an ideal in r. Let I be the subset of the ring R and let x, y ∈ I and a ∈ R.
Now we need to show that I is an ideal if and only if it satisfies: Closure under subtraction: x - y ∈ I for all x, y ∈ I, Commutativity with ring elements: a * x ∈ I and x * a ∈ I for all x ∈ I and a ∈ R. Now let us consider the steps to prove the above claim:
Closure under subtractionLet r and s be elements of ⟨r1,...,rn⟩. By the definition of ⟨r1,...,rn⟩, there are elements λ1, ..., λn and µ1, ..., µn of R such that r = λ1r1 + · · · + λnrn and s = µ1r1 + · · · + µnrn. Then r − s = (λ1 − µ1)r1 + · · · + (λn − µn)rn is again in ⟨r1,...,rn⟩.Commutativity with ring elementsLet r ∈ ⟨r1,...,rn⟩ and a ∈ R. By the definition of ⟨r1,...,rn⟩, there are elements λ1, ..., λn of R such that r = λ1r1 + · · · + λnrn. Then a · r = (aλ1)r1 + · · · + (aλn)rn is again in ⟨r1,...,rn⟩. Similarly, r · a is in ⟨r1,...,rn⟩.
Therefore, by the above closure under subtraction and commutativity with ring elements, the subset ⟨r1,...,rn⟩={λ1r1 ··· λnrn |λ1,...,λn ∈ r} is an ideal in r.
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Related Questions
1. A model rocket is launched from the roof of a building. Its flight path through the
air is modeled byn = -312 + 120 + 36, where h is the height of the rocket above the
ground in meters and tis the time after the launch in seconds.
a) What is the height of the building?
b) How long does it take for the rocket to hit the ground?
c) At what time does the rocket reach maximum height? What is the maximum
height?
d) For how long does the rocket have a height of 45m or greater?
Answers
Answer:
-156
Step-by-step explanation:
A student-athlete wants to sell hot dogs for a game. She has to pay a vendor fee of $3000 for the season & equipment will cost her $4500 for the season. Each hot dog sold will cost her $0.35. She anticipates selling 2000 hot dogs at the game.
To break even, what price should she charge for each hot dog?
Answers
To determine the price the student-athlete should charge for each hot dog to break even, we need to consider the total costs she incurs, including the vendor fee, equipment cost, and the cost per hot dog. By dividing the total costs by the anticipated number of hot dogs sold, we can find the price per hot dog that would allow her to break even.
The total costs for the student-athlete include the vendor fee of $3000 and the equipment cost of $4500, which sums up to $7500 for the season. Additionally, she incurs a cost of $0.35 per hot dog sold. With an anticipation of selling 2000 hot dogs, the total cost for the hot dogs would be 2000 x $0.35 = $700.
To break even, the total revenue from selling the hot dogs should cover the total costs. Therefore, the price per hot dog should be set in such a way that the revenue from selling 2000 hot dogs equals $7500. Dividing the total costs ($7500) by the number of hot dogs (2000), we find that the student-athlete should charge $3.75 per hot dog in order to break even. This price per hot dog would allow her to cover all her expenses and reach the break-even point.
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Laura goes to a Mexican restaurant for lunch she's planning to order either a chicken or beef dish if there are a choices for chicken and 12 choice es for beef how many options does Laura have to choose from
A. 4
B. 8
C. 12
D. 20
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plsss I'll mark Brainlest
Answers
Answer:
For class A, the test scores were closer togther so that means more students most likely scored the smae score mutiple times. I think this because the box on the box plot is not spread out like class B.
For class B, the scores varied more so the box plot has to be spread out over more values. Because of the box plot, we can see that Class B had the highest test score compared to class A.
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Solve by factoring.
f(x) = x^2 + 7x + 12
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HELP ME PLEASE!!! (20-15 points and brainlist) DO ALL QUESTIONS
Answers
Convert 43.81 ounces to grams
16 ounces = 1 pound
1 kilogram ≈ 2.2 pounds
Round your answer to the nearest hundredth
Answers
Answer:
1241.99
Step-by-step explanation:
1241.993 grams is the original answer but rounded to the nearest hundred is 1241.99
a psychologist has developed an aptitude test which consists of a series of mathematical and vocabulary problems. they want to test the hypothesis that the mean test score is 75. a random sample of 30 people have taken the test and their results recorded:
Answers
The psychologist developed an aptitude test consisting of mathematical and vocabulary problems and wants to test the hypothesis that the mean test score is 75. A random sample of 30 people took the test, and their results were recorded.
To test the hypothesis, we can perform a hypothesis test using the sample data. We will assume a null hypothesis that the mean test score is equal to 75 and an alternative hypothesis that it is different from 75. We can use a t-test for this analysis since the population standard deviation is unknown.
Using the sample data, we can calculate the sample mean and standard deviation. Let's assume the sample mean is 72.5 and the sample standard deviation is 5. Based on these values, we can calculate the test statistic using the formula:
\(\[ t = \frac{{\text{{sample mean}} - \text{{hypothesized mean}}}}{{\text{{sample standard deviation}} / \sqrt{n}}} \]\)
where n is the sample size. Substituting the values, we get:
\(\[ t = \frac{{72.5 - 75}}{{5 / \sqrt{30}}} \]\)
By calculating this value, we can determine the p-value associated with the test statistic using a t-distribution table or statistical software. If the p-value is less than the significance level (commonly 0.05), we reject the null hypothesis. If it is greater, we fail to reject the null hypothesis. This conclusion will provide evidence regarding the psychologist's hypothesis about the mean test score.
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a book cost for R150 and sold for R180 how much is the profit
Answers
Answer:
R30
Step-by-step explanation:
You would subtract R150 from R180 to get the profit.
Hope this helps :)
Geometry problem....
Answers
Answer:
Step-by-step explanation:
24/x = 28/(32 - x) Cross multiply
24 * (32 - x) = 28x Remove the brackets.
768 - 24x = 28x Add 24x to both sides
768 = 28x + 24x
768 = 52x
52x = 768 Divide by 52
x = 768/52
x = 14.77
AB = 32 - 14.77
AB = 17.23
Compute f′(a) algebraically for the given value of a. f(x)=−7x+5;a=−6
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The f′(a) when a = −6 is -7. This means that the slope of the tangent line of the graph of f(x) at x = -6 is -7.
To compute f′(a) algebraically for the given value of a, we use the following differentiation rule which is known as the Power Rule.
This states that:If f(x) = xn, where n is any real number, then f′(x) = nxⁿ⁻¹This is valid for any value of x.
Therefore, we can differentiate f(x) = −7x + 5 with respect to x using the power rule as follows:
f(x) = −7x + 5
⇒ f′(x) = d/dx (−7x + 5)
⇒ f′(x) = d/dx (−7x) + d/dx(5)
⇒ f′(x) = −7(d/dx(x)) + 0
⇒ f′(x) = −7⋅1 = −7
Hence, the derivative of f(x) with respect to x is -7.Now, we evaluate f′(a) when a = −6 as follows:f′(x) = −7 evaluated at x = −6⇒ f′(−6) = −7
Therefore, f′(a) when a = −6 is -7. This means that the slope of the tangent line of the graph of f(x) at x = -6 is -7.
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Need help asap I need to get this question right
Answers
The given expression is:
\((x-2)(x+4)\)It is required to simplify the product using the distributive property.
Apply the distributive property to the product:
\((x-2)(x+4)=x(x+4)-2(x+4)\)Apply the distributive property again to simplify the expression on the right:
\(\begin{gathered} x(x+4)-2(x+4)=x\cdot x+x\cdot4+(-2)\cdot x+(-2)\cdot4 \\ =x^2+4x-2x-8 \\ =x^2+2x-8 \end{gathered}\)The answer is x²+2x-8.
Which expression is equivalent to 7(3x+5) *
A: 10x+12
B: 3x+35
C: 21x+5
D: 21x+35
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Please make me the Brainliest!!
need help please will mark branliest
whats the perimeter of triangle
side is x+1
side is x+1
base is 2x+2
2.whats the perimeter of rectangle
length is x
width is x+3
Answers
Rectangle is 4x+6
To get perimeter of some thyroid add all the sides together
Perimeter of triangle:6x+6
Perimeter of rectangle:4x+6
passes through (-6,2) and is parallel to the line whose equation is 2x-3y=12
Answers
\(Answer:\large\boxed{y=\frac{2}{3} x+6}\)
Step-by-step explanation:
First let's convert 2x - 3y = 12 into \(y = mx + b\) form.
In order to do this, solve for y.
\(2x-3y=12\)
\(-3y=-2x+12\)
\(y=\frac{2}{3} x-4\)
This shows us that the slope is \(\boxed{\frac{2}{3}}\)
Now we use the point-slope formula:
\((y-y1)=m(x-x1)\)
where m is the slope and y1 and x1 are the point the line passes through
Using the point (-6,2) and slope, 2/3, we can find the equation:
\((y-2)=\frac{2}{3} (x-(-6))\)
\((y-2)=\frac{2}{3} (x+6)\)
\((y-2)=\frac{2}{3} x+4\)
\(\large\boxed{y=\frac{2}{3} x+6}\)
the first several terms of a sequence {an} are: 14,−19,114,−119,124,.... assume that the pattern continues as indicated, find an explicit formula for an.
Answers
The explicit formula for the sequence {an} is:
an = 14 + 5n + 100(n-1) for even n
an = -(14 + 5n + 100(n-1)) for odd n
To find the explicit formula for the sequence {an} with the given terms 14, -19, 114, -119, 124, ...,
Step 1: Observe the pattern
We can see that the signs alternate between positive and negative, and the absolute values of the terms follow the pattern 14, 19, 114, 119, 124, ...
Step 2: Identify the explicit formula
The absolute values can be expressed as the sequence: 14, 14 + 5, 14 + 100, 14 + 105, 14 + 200, ...
This suggests a pattern: an = 14 + 5n + 100(n-1) when n is even, and an = -(14 + 5n + 100(n-1)) when n is odd.
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A tank is full of water when a valve at the bottom of the tank is opened. The equation V = 62(151 – t) gives the volume of water in the tank, in cubic meters, after t hours. What is the volume of water in the tank before the valve is opened? cubic meters How long does it take the tank to fully empty? hours Find an equation for dv/dt
dv/dt= Preview What is the flow rate after 23 hours? Select an answer When is the water flowing out of the tank the fastest? t= hours
Answers
The volume of water in the tank before the valve is opened can be found by plugging in t = 0 into the given equation V = 62(151 - t), which gives the initial volume.
To find the volume of water in the tank before the valve is opened, we plug in t = 0 into the given equation: V = 62(151 - t). Substituting t = 0 gives V = 62(151), which simplifies to V = 9338 cubic meters.
To find how long it takes the tank to fully empty, we set V = 0 in the equation V = 62(151 - t) and solve for t. This gives 0 = 62(151 - t), which simplifies to 151 - t = 0. Solving for t gives t = 151 hours.
The equation dv/dt represents the rate at which the volume of water is changing with respect to time. To find the flow rate after 23 hours, we substitute t = 23 into the derivative equation dv/dt. This gives dv/dt = 62.
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Naomi plotted the graph below to show the relationship between the temperature of her city and the number of popsicles she sold daily:
A scatter plot is shown with the title Naomis Popsicle Stand. The x axis is labeled High Temperature, and the y-axis is labeled Number of Popsicles Sold. Data points are located at 90 and 20, 85 and 17, 70 and 14, 75 and 20, 60 and 16, 50 and 14, 60 and 12, 40 and 10, 50 and 12, 80 and 8.
Part A: In your own words, describe the relationship between the temperature of the city and the number of popsicles sold. (2 points)
Part B: Describe how you can make the line of best fit. Write the approximate slope and y-intercept of the line of best fit. Show your work, including the points that you use to calculate the slope and y-intercept.
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Answer: Part A: The graph shows a negative relationship between the temperature of the city and the number of popsicles sold. As the temperature increases, the number of popsicles sold decreases.
Part B: To make the line of best fit, we need to draw a line that comes closest to the data points. We can find the slope and y-intercept of the line by using two of the data points on the graph. Let's choose the points (90, 20) and (40, 10).
Slope = (change in y) / (change in x) = (20 - 10) / (90 - 40) = 0.2
To find the y-intercept, we can use the slope-intercept form of a line: y = mx + b, where m is the slope and b is the y-intercept. Let's use the point (90, 20) and plug in the slope we just found:
20 = 0.2(90) + b
b = 2
Therefore, the approximate slope of the line of best fit is 0.2 and the approximate y-intercept is 2.
Step-by-step explanation:
What to use? Pick two points from the table and
use the slope formula to find the slope.
Example: The table below has a constant rate of
change. Find the slope.
Х
Y
- 1
4
0
6
1
8
2
10
3
12.
4.
14
Answers
Answer:
The slope is 1/2
Step-by-step explanation:
AS seen in the graph I provided you can see that the points are increasing by 1/2
a line has a slope of 8 and includes the points (-7,10) and (-9,h). what is the value of h
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Answer:
- 6---------------------
Use the slope equation:
m = (y₂ - y₁) / (x₂ - x₁), where m- slopeSubstitute given values for the slope and coordinates and and solve for h:
8 = (h - 10) / (- 9 - (- 7))8 = (h - 10) / (- 2)-16 = h - 10h = -16 + 10h = - 6The value of h is - 6.
Write the equation y= 4/3 x + 2 in standard form also please Include step by step process
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The standard form of the equation of the line
The equation of the line can be expressed in several forms. One of them is the standard form:
Ax + By = C
Where A, B, and C are constants, and x and y are the variables.
We have the equation:
\(y=\frac{4}{3}x+2\)Multiplying by 3:
\(3y=4x+6\)Subtracting 4x:
\(-4x+3y=6\)Is the required standard form of the line
A local fair has an entrance fee of $10. Each ride costs $1.25.
a. If Todd has $26.25, write an equation and determine how many rides could he ride
after he pays the entrance fee?
b. Explain the steps in solving the equation by naming the property or properties you used.
c. What mathematical symbol represents the word “is”?
d. What is the first step in solving this problem?
(Please answer as quick as YOU can!)
Answers
Answer:
Step-by-step explanation:
A. 26.25<=1.25x+10 because he can spend no more than 26.25
C. The '=' sign
D solve for X. Subtract 10 from both sides. Then divide both sides by 1.25.
Find the exact value of sin 7 in simplest form.
4
U
7
√33
T
Answers
Sin 7° is equal to sin(0.1221) = 0.1218693.
How to convert degree to radians ?We must multiply the supplied value by \(\pi\)/180 to convert the angle's degree value to its counterpart in radians.
According to the given information
The angle 7°, or sin 7 degrees, is located between 0° and 90°. (First Quadrant).
Sin 7° value = 0.1218693 because the sine function is positive in the first quadrant.
Sin 7 degrees has a numeric value of 0.121869343.
Another way to define sin 7 degrees is to use the radian equivalent of the angle (7 degrees) (0.12217 . . .).
Using the degree to radian conversion, we can see that 7 degrees equals 7° (\(\pi\)/180°) in radians. rad = 0.1221 . . .
Sin 7° is equal to sin(0.1221) = 0.1218693.
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Question 10 part a and b please I need help due tmmr
Answers
Answer:
∠ E ≈ 38.9° , ∠ G ≈ 79.6°
Step-by-step explanation:
given 3 sides of a triangle, to find any of the angles use the Cosine rule
(A)
cos E = \(\frac{f^2+g^2-e^2}{2fg}\)
where f is the fide opposite ∠ F , g is the side opposite ∠ G and
e is the side opposite ∠ E
here f = EG = 42 , g = EF = 47 and e = FG = 30 , then
cos E = \(\frac{42^2+47^2-30^2}{2(42)(47)}\) = \(\frac{1764+2209-900}{3948}\) = \(\frac{3073}{3948}\) , then
∠ E = \(cos^{-1}\) ( \(\frac{3073}{3948}\) ) ≈ 38.9° ( to the nearest tenth )
(b)
similarly, using e, f , g as above in part A
cos G = \(\frac{e^2+f^2-g^2}{2ef}\) = \(\frac{30^2+42^2-47^2}{2(30)(42)}\) = \(\frac{900+1764-2209}{2520}\) = \(\frac{455}{2520}\) , then
∠ G = \(cos^{-1}\) ( \(\frac{455}{2520}\) ) ≈ 79.6° ( to the nearest tenth )
A bag contains 6 red marbles and 1 blue marble. A marble is taken at random, put to one side, and then another marble is taken at random. What is the probability that at least one of the marbles takes was blue?
Give your answer as a fraction in its simplest form
Answers
We have 6 red and 1 blue marble thus the probability of drawing blue marble = 1/7
To understand probability as a concept, pay attention to the steps below.
Step 1. Multiply the individual probabilities to obtain the chance of numerous separate events.
Step 2. As there are two separate events in this scenario, double the probabilities of each.
Step 3. Add the individual probabilities to obtain the chance of several events that are mutually exclusive.
In this bag of 7 marbles, there is 1 blue one. Assume that each is marked with a number. Choosing blue-1 has a 1/7 chance of happening. (Why? As there are 7 marbles that may be chosen, each with an equal probability, and since those 7 occurrences are mutually exclusive, the 7 probabilities total up to 1.)
The probability of choosing blue-2 is similarly 1/7; the same goes for blue-3,..., and blue-8. To determine the likelihood of picking a blue, add those up (and blue).
Step 4. Do the same for red next.
Thus the probability of drawing blue marble = 1/7
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The table shows the relationship between time spent running and distance traveled. A 2-column table with 5 rows. The first column is labeled time (minutes) with entries 1, 2, 3, 4, 5. The second column is labeled distance (feet) with entries 530; 1,050; 1,600; 2,110; 2,650. Which type of model best describes the relationship? linear, because the r value for the linear model is closest to 1exponential, because the r value for the exponential model is closest to 0linear, because the rate of change between each pair of points is exactly 520exponential, because the rate of change between each pair of points is 1. 98.
Answers
The r value for the linear model is closest to 1.
We have two columns in this question, one for you to enter the time and the other for the distance. We must determine whether the relationship is exponential or linear.
I'll start by explaining the differences between linear and exponential relationships, after which we'll observe the data and find the answer, which is as follows:
linear, as the linear model's r value, is the closest to 1.
exponential and linear data
It is linear if close numbers are assumed for the absolute value of the change.
If so, the rate is exponential.
Data:
t = 1 -> d = 530
t = 2 -> d = 1050
t = 3 -> d = 1600
t = 4 -> d = 2110
t = 5 -> d = 2650
Subtractions:
2650 - 2110 = 540
2110 - 1600 = 510
1600 - 1050 = 550
1050 - 530 = 520
Divisions(rates):
2650/2110 = 1.25
2110/1600 = 1.32
1600/1050 = 1.52
1050/530 = 1.98
The rates change considerably more than the subtractions, so it is a linear relationship, and the correct option is: linear because the r value for the linear model is closest to 1.
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Given two points, how many different planes pass through the two points?
Answers
Answer:
an infinite number of planes
Step-by-step explanation:
i looked it up
A number is decreased by seven. Then, the new number is multiplied by 3 to get an answer of -9. What is the original number? ( please hurry i need to finish my assesment )
A.) - 4
B.) - 30
C.) 4
D.) 30
Answers
Answer: -4
Step-by-step explanation:reverse the steps
3(x-7)= -9
3(x-7)/3= -9/3
x-7= -3
x= -3+7
x= -4
please please help it’s a rest
Answers
hellppp me pleaseee !
