Answer: B
Step-by-step explanation:
The pumpkin to green bean ratio is 2:1
That means x, the amount of green beans, has to be 1/2 the amount of pumpkin. Therefore the amount of pumkin is 2x.
The total amount of cans would be 3x, which is 114
3x = 114
114/3 = x
x = 38 cans of green beans
there are 8 people who would like to share a box of granola bars that contains 8 bars haw many granola bars does each person get if they share equally
Answer:
One bar each
Step-by-step explanation:
8 people/8granola bars =1 bar each
A financial analyst following Singh, Inc. has created the following probability distribution to describe Singh's expected return over the next year. Singh's fortunes rely on the success of a new product. The probability distribution shown reflects the analyst's expectation of the stock's return depending on how well the new product sells. What is the expected return on the stock? 7.50% 2.00% 22.50% 19.00% 14.00% What is the expected standard deviation of stock returns? 30.51% 26.57% 36.82% 24.23% 31.21%
The expected return on the stock is 22.50% and the expected standard deviation of stock returns is 26.57%
The standard deviation is a measure of the spread of a set of data around its mean (average) value. It is a statistical measure of the dispersion of data. The standard deviation is the square root of the variance, which is the average of the squared differences between each data point and the mean.
The expected return on the stock is the weighted average of the possible returns, with the weights being the probabilities of those returns occurring. To calculate the expected return, we multiply each return by its corresponding probability and add the products together.
Expected return = (7.50% * 0.2) + (2.00% * 0.1) + (22.50% * 0.3) + (19.00% * 0.1) + (14.00% * 0.3) = 7.50% + 0.2% + 6.75% + 1.90% + 4.20% = 22.50%
The expected standard deviation of stock returns is a measure of the volatility or risk of the stock. It is calculated by taking the square root of the variance, which is the average of the squared differences between each return and the expected return.
Expected standard deviation = √(((7.50% - 4.20%)^2 * 0.2) + ((2.00% - 4.20%)^2 * 0.1) + ((22.50% - 4.20%)^2 * 0.3) + ((19.00% - 4.20%)^2 * 0.1) + ((14.00% - 4.20%)^2 * 0.3)) = √(26.57%)
So, the expected return on the stock is 4.20% and the expected standard deviation of stock returns is 26.57%.
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Sheila and Dora worked 3 1 2 hours and 5 1 2 hours, respectively, on a programming project. They were paid $198 for the project. How much (in dollars) did each earn?
Sheila get $77 and Dora get $121 paid.
What is Fraction?The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.
The number of parts into which the whole has been divided is shown by the denominator. It is positioned in the fraction's lower portion, below the fractional bar.How many sections of the fraction are displayed or chosen is shown in the numerator. It is positioned above the fractional bar in the upper portion of the fraction.Given:
Sheila and Dora worked 3 1/2 hours and 5 1/2 hours.
Total they get paid = $198.
In one hour the amount of work they done is 7/2 and 11/2.
So, the ratio of their work = (7/2)/ (11/2)= 7/11
So, the amount Sheila get
= 7/18 x 198
= $77
and, the amount Dora get
= 11/18 x 198
= $121
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From a recent study, it was found that 10% of students travel outside of the country during spring break. Is this an example of descriptive statistics or inferential statistics?
This is an example of descriptive statistics. Descriptive statistics is used to summarize, organize, and present data in a meaningful way.
Is this an example of descriptive statistics or inferential statistics?It involves the collection, analysis, interpretation, and presentation of data. In this case, the study found that 10% of students travel outside of the country during spring break. This is a summary of the data collected and presents it in a way that is easy to understand. It gives a percentage of students who travel outside of the country during spring break, which can be used to describe the behavior or characteristics of students.On the other hand, inferential statistics is used to make predictions or draw conclusions about a population based on a sample of data. It involves using statistical methods to make inferences about a population from a sample. For example, if a study found that 10% of students at a particular university travel outside of the country during spring break, inferential statistics would be used to make predictions or conclusions about the behavior or characteristics of students at other universities.To learn more about inferential statistics refer to:
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Please help
The division 7+5i/2-5i, is performed by multiplying the numerator and denominator by
View image
Which of the following sets is closed under division?A. Natural numbersB. Non zero intergersC. Irrational numbersD. Non zero rational numbers
A set is considered closed under a particular operation if the result of that operation applied to any two elements of the set is also an element of the set.
Division is an operation on two elements, where one element is divided by the other.
A. Natural numbers are not closed under division since when we divide two natural numbers, the result may not be a natural number. For example, when we divide 4 by 2, the result is 2, which is a natural number. However, when we divide 6 by 3, the result is 2.00, which is not a natural number.
B. Non zero intergers are not closed under division since when we divide two non zero integers, the result may not be a non zero integer. For example, when we divide 8 by 2, the result is 4, which is a non zero integer. However, when we divide 6 by 3, the result is 2.00, which is not a non zero integer.
C. Irrational numbers are not closed under division since when we divide two irrational numbers, the result may not be an irrational number. For example, when we divide √2 by √2, the result is 1, which is an irrational number. However, when we divide √3 by √2, the result is 1.73, which is not an irrational number.
D. Non zero rational numbers are closed under division since when we divide two non zero rational numbers, the result will always be a non zero rational number. For example, when we divide 6 by 3, the result is 2, which is a non zero rational number. Similarly, when we divide 8 by 4, the result is 2, which is also a non zero rational number.
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What would the cash discount statement "3/10, 2/20, net/30" mean on an
invoice?
Cash discounts describe a reward that a seller provides to a buyer in exchange for paying a bill in full before the due date.
Is cash discount recorded in income statement?In contrast to trade discounts, which are recorded in buy and sales books, cash discounts are an expense to a corporation that appear in the profit and loss account.Using the price and the rate, the formula determines the cash discount. Cash Discount is calculated as Purchase Price x Discount Rate. For instance, if a product costs $200 and a 10% discount is applied, the cash discount would be $20, translating to a $20 savings for the customer.The discount that the seller of a product provides to the buyer at the time of payment for the purchase is referred to as a cash discount. This discount is offered at the amount of the invoice.To learn more about discounts refer to:
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Haley solves the inequality - 13 ≥r+7 and
graphs the solution on a number line with a solid circle at -20 and an
arrow pointing left. Is she correct? Support your answer, and give
the correct description if she is incorrect.
Haley is right because the solution is r ≤ -20
What is an equation?An equation is an expression that shows the relationship between numbers and variables.
Inequalities is used for the non equal comparison of these numbers and variables.
Given the inequality:
-13 ≥ r + 7
Add 13 to both sides:
-13 + 13 ≥ r + 7 + 13
0 ≥ r + 20
subtract r from both sides:
0 - r ≥ r + 20 - r
-r ≥ 20
divide both sides by -1:
r ≤ -20
The solution is a graph on a number line with a solid circle at -20 and an arrow pointing left
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9.) Multiple Choice: In triangle ABC, side AB has a length of 6cm and side BC has a length of 10cm. What is the length of AC?
The length of AC of the given right angle triangle is; AC = 8
How to use Pythagoras theorem?Pythagoras Theorem is defined as the way in which you can find the missing length of a right angled triangle. The triangle has three sides, the hypotenuse, Opposite and the adjacent side.
Now, the given triangle is a right angle triangle and using Pythagoras theorem, we can say that;
AC = √(BC² - AB²)
Thus;
AC = √(10² - 6²)
AC = √64
AC = 8
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Fix the inequality below to correctly show the domain of
the function.
The domain of the graph will be [-1.50, 1). Then the domain in the form of inequality will be - 1.50 ≤ x < 1.
How to draw the graph of the function?The collection includes all locations on the surface of the shape (x, f(x)) that make up a function of f's graph. We may alternatively say that the graph of f is the curve of y = f. (x). As a result, the diagram of an equation is a particular instance of the graphs of functions.
The domain of the graph is given as,
- 1.50 ≤ x < 1
The domain of the graph will be [-1.50, 1). Then the domain in the form of inequality will be - 1.50 ≤ x < 1.
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The complete question is attached below.
In which sentence does the verb most
accurately and vividly describe the
action?
A. The joke was so hilarious that the audience roared!
B. The joke was so hilarious that the audience smirked!
C. The joke was so hilarious that the audience smiled!
Sentence A: "The joke was so hilarious that the audience roared!" most accurately and vividly describes the action.
Which sentence does the verb most accurately and vividly describe the action?In this sentence, the verb "roared" is used to describe the reaction of the audience to the joke. Roaring is a strong and lively verb that conveys the idea that the audience was highly entertained and had a strong emotional reaction to the joke.Sentence B: "The joke was so hilarious that the audience smirked!" is not vivid enough to describe the action as the audience reaction is not clearly defined. Smirking is usually a subtle expression that might be difficult to interpret.Sentence C: "The joke was so hilarious that the audience smiled!" is also not vivid enough to describe the action as smiling is a common reaction to something funny and it doesn't clearly convey the level of audience's reaction.Overall, Sentence A is the most accurate and vivid as it describes the audience's reaction in a lively and emotional way.To learn more about verb refer to;
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Sentence A is the most accurate and vivid as it describes the audience's reaction in a lively and emotional way.
Which sentence does the verb most accurately and vividly describe the action?In this sentence, the verb "roared" is used to describe the reaction of the audience to the joke. Roaring is a strong and lively verb that conveys the idea that the audience was highly entertained and had a strong emotional reaction to the joke.
Sentence B: "The joke was so hilarious that the audience smirked!" is not vivid enough to describe the action as the audience reaction is not clearly defined. Smirking is usually a subtle expression that might be difficult to interpret.
Sentence C: "The joke was so hilarious that the audience smiled!" is also not vivid enough to describe the action as smiling is a common reaction to something funny and it doesn't clearly convey the level of audience's reaction.
Overall, Sentence A is the most accurate and vivid as it describes the audience's reaction in a lively and emotional way.
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Tennis great Roger Federer made 63% of his first serves in a recent season. When Federer made his first serve, he won 78% of the points. When Federer missed his first serve and had to serve again, he won only 57% of the points. Suppose you randomly choose a point on which Federer served. You get distracted before seeing his first serve but look up in time to see Federer win the point. What’s the probability that he missed his first serve?
The probability that Federer missed his first serves given that he won the point is 0.30%.
What is the probability that he missed his first serve?Generally, To solve this problem, we can use Bayes' theorem. Bayes' theorem states that:
P(A|B) = P(B|A) * P(A) / P(B)
where
A and B are events, and
P(A|B) is the probability of event A given that event B has occurred.
In this problem, let A be the event that Federer missed his first serve, and let B be the event that he won the point.
We are trying to find P(A|B), the probability that Federer missed his first serve given that he won the point.
We know that:
P(A) = 1 - 0.63 = 0.37 (the probability that Federer missed his first serve, given that he made 63% of his first serves)
P(B|A) = 0.57 (the probability that Federer won the point given that he missed his first serve)
P(B|A') = 0.78 (the probability that Federer won the point given that he made his first serve)
We can use these probabilities and Bayes' theorem to find the answer:
P(A|B) = P(B|A) * P(A) / ( P(B|A) * P(A) + P(B|A') * P(A'))
P(A|B) = 0.57 * 0.37 / ( 0.57 * 0.37 + 0.78 * 0.63)
= 0.30029 or 30.33%
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Answer:
0.3
Step-by-step explanation:
Tennis great Roger Federer made 63% of his first serves in a recent season. When Federer made his first serve, he won 78% of the points. When Federer missed his first serve and had to serve again, he won only 57% of the points. Suppose you randomly choose a point on which Federer served. You get distracted before seeing his first serve but look up in time to see Federer win the point. What’s the probability that he missed his first serve?
In a classroom, 100 plastic fish are in a tub. The tub is hidden from the student view. Some fish are green, and the rest are yellow. The students know that there are 100 fish, but they don’t know how many of each color there are. The students go fishing, each time, picking a random fish from the tub, recording its color, and throwing the fish back in the tub. At the end of the day, 65 fish have been chosen, 12, green and 53 yellow. What is the best estimate you can give for the number of green fish in the number of yellow fish in the tub? Describe how to calculate this best estimate, and explain why your method of calculation makes sense in a way that a seventh grader my understand. Is your best estimate necessarily accurate? Why or why not?
The best estimate for the number of fish is that there are 82 yellow fish and 18 green fish, which you can calculate using the rule of three. This estimate is expected to be accurate.
What is the estimated number of fish of each color?This can be calculated using a rule of three as follows. Let's start with the yellow fish:
53 = 65
x = 100
53 x 100 / 65 = 82 yellow fish
Green fish:
12 = 65
x = 100
12 x 100 / 65= 18 green fish
Is this estimate accurate?This estimate is expected to be accurate; however, there might be small differences in real life such as 80 yellow fish and 20 green fish.
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propose five combinations of aqueous ionic reagents that likely would form a precipitate when they are mixed. Write the balanced full molecular equation and the balanced net ionic equation for each of your choices.
Equations will be AgNO3(aq)+HCl(aq)⟶AgCl(s)+HNO3(aq)
AgNO3+HCl⟶AgCl+HNO3 for acid
Ag:+1⟶+1
NO3:−1⟶−1
H:+1⟶+1
Cl:−1⟶−1
According to the solubility table, nitrates are always soluble, so the strong ionic bond between silver ions and nitrate ions are broken by water molecules because of ion-dipole attraction. Acids ionize in water to give mobile ions, so hydrogen chloride in aqueous solution gives out hydrogen ions (and form hydronium ions) and chloride ions.
There are five types of particles in the solution :
H2O molecules, Ag+ ions, NO3− ions, H+ ions, and Cl− ions. (Slight ionization of water is neglected in this case.) They freely bump into each other as they are mobile. However, this freely moving condition is inhibited by the interaction between Ag+ ions and Cl− ions.
According to the solubility table, AgCl is insoluble in water. When Ag+ ions and Cl− ions bump into each other, they strongly attract each other, in which the strong ionic force cannot be separated by the ion-dipole force between them and H2O molecules. As a result, AgCl evolves as a white solid.
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1) Solve and graph the inequalitiy x + 8 < - 3
2) Write a statement that repersent the number line below
the pet shelter, for every 7 dogs there are 5 cats. What is the ratio of dogs to cats?
a) The inequality equation is x < -11
b) The inequality equation is x < 7
c) The ratio of dogs to cats is 7 : 5
What is an Inequality Equation?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.
Given data ,
Let the inequality equation be represented as A
Now , the value of A is
a)
Substituting the values in the equation , we get
x + 8 < -3 be equation (1)
On simplifying the equation , we get
Subtracting 8 on both sides of the equation , we get
x < -11
Therefore , the inequality is x < -11
b)
The inequality representing the number line is x < 7
c)
Let the ratio be represented as A
The number of dogs = 7
The number of cats = 5
The ratio of dogs to cats = number of dogs / number of cats
Substituting the values in the equation , we get
The ratio of dogs to cats = 7 / 5
Therefore , the ratio is 7 : 5
Hence , the inequality relation is solved
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Factor the expression 48+72 using the GCF
Using the greatest common factor the expression is factored to be
12 (4 + 6)What is greatest common factor?The largest number among all the common factors of the given numbers is known as the GCF (Greatest Common Factor) of two or more numbers.
The greatest integer that may be used to divide two natural numbers x and y without producing any remainders is called their GCF.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
The greatest common factor of 48 and 72 is 12
hence we have
12 (4 + 6)
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identify the sampling method that was used. fifteen instructors are chosen using random numbers to implement a new learning strategy.
Random sampling is the sampling method that was used.
Random sampling is a method in which all members of the population have an equal chance of being selected for the sample. In this case, fifteen instructors were chosen using random numbers to implement the new learning strategy.Random sampling is a useful tool when a researcher needs to select a representative sample from a population. By using random numbers to choose the sample, the researcher can ensure that the sample is truly representative of the population. Random sampling also helps to reduce bias, as the researcher has no control over who is chosen for the sample.
To calculate the probability of being chosen for the sample, the formula is (1/n) x 100, where n is the total number of population members. In this case, the population was the group of all instructors, and the sample was the group of fifteen instructors chosen. Therefore, the probability of being chosen is (1/n) x 100 = (1/15) x 100 = 6.67%.
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If m
and mzABD = (4x + 14)°. Find the
measurements of CBD and
The measure of the angles <ABC and <CBD are 48⁰ and 42⁰
How to calculate the measure of angles?The angle ABC and angle CBD are complementary angles, m <ABC=(3x+6)⁰ and m<CBD=(4x-14)⁰
The angles are complementary which means that they sum to 90⁰
Therefore, we get: <ABC + <CBD = 90⁰
3x + 6 + 4x - 14 = 90 ------ Substituting 3x + 6 and 4x - 14 for < ABC and <CBD
7x = 98
x= 98/7 = 14⁰
Therefore; Measure of <ABC:
Measure of<CBD:
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Correct question:
Find the measure of each angle angle ABC and angle CBD are complementary angles, m angle ABC=(3x+6)degrees, and m angle CBD=(4x-14)degrees stuck on this problem, Thank you fo
Ben scores 56 out of 79 on a maths test
what percent did he get?
If Ben scores 56 out of 79 on a maths test. The percentage he get is: 0.7089.
How to find the percentage he got?Using this formula to determine the percentage he got
Percentage gotten = Number of scores / Total score
Where:
Number of score = 56
Total score = 79
Let plug in the formula
Percentage gotten = 56 ÷ 79
Percentage gotten = 0.70886
Percentage gotten = 0.7089 (Approximately)
Therefore we can conclude that he got 0.7089 .
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Need help with #6 please help
Answer:
20 m
Step-by-step explanation:
division story for 40 divided by 4
I have 40 cupcakes in my house. That day, 3 friends came over. All of us have the same number of cupcakes each. How many cupcakes did we each eat?
The story of 40 divided by 4 reminding everyone about the importance of fairness and sharing.
There were 40 colorful candies waiting to be shared equally among 4 friends.
The candies were so delicious and enticing that the friends couldn't wait to indulge in their sweet treat.
The four friends, named Alex, Ben, Chris, and Dana, gathered around a table with their empty candy bags in front of them.
They were filled with excitement as they knew it was time to divide the candies fairly.
With the candies in the center of the table, they decided to take turns dividing them into smaller piles.
Alex went first and carefully divided the candies into 10 piles, making sure each pile had an equal number of candies.
This left 10 candies in each pile.
Then, it was Ben's turn. He took one pile of 10 candies and divided it into two equal piles, with 5 candies in each.
Ben did the same with the other nine piles, resulting in a total of 20 piles, each containing 5 candies.
Next, it was Chris's turn. He picked one pile with 5 candies and divided it into two equal piles, leaving 2 candies in each pile.
Chris repeated this process with the remaining 19 piles, resulting in a total of 40 piles, each containing 2 candies.
Finally, it was Dana's turn. Dana took one pile with 2 candies and divided it equally between two empty candy bags, leaving 1 candy in each bag. Dana repeated this process with the remaining 39 piles, resulting in a total of 80 candy bags, each containing 1 candy.
In the end, all four friends had received their fair share of candies. They joyfully took their candy bags and started savoring the delicious treats, grateful for their successful division.
And so, the story of 40 divided by 4 came to a sweet conclusion, reminding everyone about the importance of fairness and sharing.
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Sauv drove 296 miles in 8 hours.If he continued at the same rate,how far would he travel in 10 hours?
Given the information below, match the step with the reason it can be stated.
The reasons that matches each steps are:
1. Given (first occurrence)
2. Given (second occurrence)
3. Vertical angles theorem
4. ASA
5. CPCTC
What is the CPCTC?CPCTC stands for "corresponding parts of congruent triangles are congruent", which connote that if two triangles are congruent, then every parts that are corresponding will be congruent to each other also.
Thus the reason that match each steps would be stated as follows:
1. C is the intersection point of CD and EB » Given (first occurrence)
2. AC ≅ EC and <A ≅ <E » Given (second occurrence)
3. <ACB ≅ <ECD » Vertical angles theorem
4. ∆ABC ≅ ∆EDC » ASA
5. AB ≅ ED » CPCTC
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What is 2+2
PLEASE HELP!!!!
Answer:
4
Step-by-step explanation:
2+2=4
NO LINKS!!
44. Between which two integers would the solution to 4^x = 50 lie? Second, find the exact solution by changing to logarithmic form and solving.
Answer: To find the two integers between which the solution to 4^x = 50 lies, we can take the logarithm base 4 of both sides of the equation: x = log4(50). The logarithm of 50 with base 4 is approximately 2.3219. Since the logarithm of a number is always positive, the solution to 4^x = 50 lies between 2 and 3.
To find the exact solution, we can change the equation to logarithmic form by taking the logarithm base 4 of both sides: log4(4^x) = log4(50)
Using the logarithmic identity: log(b^x) = x*log(b)
So we have: x*log(4) = log(50)
Dividing both sides by log(4), we have:
x = log(50) / log(4)
To find the exact solution we can approximate log(50) / log(4) using a calculator or logarithm tables. The exact solution is x ≈ 2.32192809488736218170856773213.
So, the exact solution is x = 2.32192809488736218170856773213, which is between 2 and 3, as stated before.
Step-by-step explanation:
The deck of a bridge is suspended 285 feet above a river. If a pebble falls off the side of the bridge, the height, in feet, of the pebble above the water surface after t seconds is given by
y = 285 − 16t2.
(a). Find the average velocity (in ft/s) of the pebble for the time period beginning when t = 2 and lasting the following amount of time.
(I) 0.1 seconds = ___ ft/s (ii) 0.05 seconds = _____ft/s (iii) 0.01 seconds = _____ ft/s (b). Estimate the instantaneous velocity (in ft/s) of the pebble after 2 seconds. = ____ft/s
(a) (i) Average velocity = -160 ft/s, (ii) Average velocity = -320 ft/s, (iii) Average velocity = -1600 ft/s. (b) Instantaneous velocity = -320 ft/s.
(a) (i) Average velocity = change in height/change in time
= (285 - (285 - 16(2.1)^2))/0.1
= -160 ft/s
(ii) Average velocity
= (285 - (285 - 16(2.05)^2))/0.05
= -320 ft/s
(iii) Average velocity
= (285 - (285 - 16(2.01)^2))/0.01
= -1600 ft/s.
(b) Instantaneous velocity = -320 ft/s.
The average velocity of the pebble after t seconds is given by the equation y = 285 − 16t2. To find the average velocity for the time period beginning when t = 2, we can calculate the change in height of the pebble over the given time period and divide it by the change in time. For instance, for 0.1 seconds, we can calculate the average velocity by dividing the change in height (285 - (285 - 16(2.1)^2)) by the change in time (0.1). This gives us an average velocity of -160 ft/s. Similarly, for 0.05 and 0.01 seconds, the average velocity can be calculated to be -320 ft/s and -1600 ft/s respectively. To estimate the instantaneous velocity of the pebble after 2 seconds, we can assume that the change in time is very small and approximate the velocity to be the average velocity for 0.01 seconds. This gives us an instantaneous velocity of -320 ft/s.
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Mrs. Gaither had 22 students in her
first class of the day. Today, only 18 of
them were present. What fraction of
her students were ABSENT? Write the
fraction in simplest terms.
I NEED IT ASAP PLEASE
Answer:
2/11
Step-by-step explanation:
Normally 22 kids. 22 kids on the attendance roster.
18 kids say, "Here!" when their name is called.
22 - 18 is 4. 4 kids don't say anything when their name is called because they are at home still. 4 kids absent.
4 out of 22 kids absent. 4/22 kids absent.
4/22 simplifies to 2/11.
2/11 of the kids are absent.
show that if a and b both have linearly independent column vectors, then the column vectors of c will also be linearly independent.
To show that if the column vectors of matrix A and B are linearly independent, then the column vectors of matrix C = A*B will also be linearly independent, we can use the following proof:
Assume that the column vectors of matrix A and B are linearly independent. Let's assume that the column vectors of matrix C = A*B are linearly dependent.
This means that there exists a non-trivial linear combination of the column vectors of C that results in the zero vector.
C = AB => c_1 = Ab_1, c_2 = Ab_2, ... c_n = Ab_n
Let's assume that a linear combination of c1, c2, ..., cn results in the zero vector.
a_1c_1 + a_2c_2 + ... + a_n*c_n = 0
Now, we can substitute the matrix multiplication for ci:
a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n = 0
Since matrix A is non-singular, we can left multiply by A^-1
A^-1*(a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n) = A^-1*0
This simplifies to
a_1b_1 + a_2b_2 + ... + a_n*b_n = 0
This is a non-trivial linear combination of the column vectors of matrix B that results in the zero vector, which contradicts the assumption that the column vectors of matrix B are linearly independent. Therefore, the column vectors of matrix C = A*B must also be linearly independent.
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11. What equation in slope-intercept form represents the line that passes through the points (-3, 4) and (0, -1)?
Slope is a metric that expresses how inclined a line is with respect to the horizontal.Answer is y = (-5/3)x - 1 .
What is the equation ?Slope is a metric that expresses how inclined a line is with respect to the horizontal.
The anti-clockwise measured tangent of the angle between a line and the positive x-axis is the slope of a straight line.
The slope of the line will be m = tan if is the angle between a straight line and the positive x-axis.
Y/X = (Y2 - Y1)/(X2 - X1) = m = tan
A straight line's equation is (y - y1)/(x - y1) = (y2 - y1)/(x2 - x1) if it goes through the points (x1, y1) and (x2, y2).
Y = mx + c is the equation of a straight line in slope intercept form.
Solution:
Given: A straight line connects (-3, 4) and (0, -1).
To determine: The straight line equation
Since ,
Since (-3,4) and (0,-1) are on a straight line, its equation will be as follows:
y - y1 / x - x1 = y2 - y1 / x2 - x1
→ (y - 4)/(x - (-3)) = (-1 - 4)/(0 - (-3))
→ (y - 4)/(x + 3) = - 5/3
→ y - 4 = (-5/3)(x + 3)
→ y - 4 = (-5/3)x - 5
→ y = (-5/3)x - 5 + 4
→ y = (-5/3)x - 1
Hence ,
The necessary formula is y = (-5/3)x - 1.
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show that there are exactly labeled simple graphs on n vertices. for any value of , how many of those labeled simple graphs have exactly edges?
It is proved that there are exactly [tex]2^{\frac{n(n-1)}{n}}[/tex] labeled simple graphs on n vertices. And about [tex]\left(\begin{array}{c}{ \frac{n(n-1)}{2}}&\\m&\end{array}\right)[/tex] labeled simple graphs that have exactly m edges.
(i) The number of edges that can be on n vertices is [tex]\frac{n(n-1)}{2}[/tex]. We have (n-1) choices to put an edge for each vertex. Also, we have n choices to select the first vertices. So in total, we have n(n-1) choices but each edge is counted twice. Then, all distinct edges on n vertices are given by [tex]\frac{n(n-1)}{2}[/tex].
Let us consider H to be any graph that lies on n vertices. And this denotes all the edges [tex]e_1, e_2, e_3,\cdots \cdots \cdots e_{\frac{n(n-1)}{2}}[/tex]. For each edge [tex]e_i[/tex], the number of choices we have is two. One is [tex]e_i[/tex] is an edge in H. Another is [tex]e_i[/tex] is not an edge in H. This is therefore written as 2×2×..............×2 choices, and this number [tex]2^{\frac{n(n-1)}{n}}[/tex] equals labeled simple graphs on n vertices.
(ii) Out of [tex]\frac{n(n-1)}{2}[/tex] edges, we can select m edges for a graph by [tex]\left(\begin{array}{c}{ \frac{n(n-1)}{2}}&\\m&\end{array}\right)[/tex]ways. Then, the number of graphs that are present on n vertices with m edges is written as [tex]\left(\begin{array}{c}{ \frac{n(n-1)}{2}}&\\m&\end{array}\right)[/tex].
The complete question is -
(i) Show that there are exactly [tex]2^{\frac{n(n-1)}{n}}[/tex] labeled simple graphs on n vertices.
(ii) For any value of, how many of those labeled simple graphs have exactly m edges?
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