A, B and C are coordinates on the line y = x - 6. Use the table of values to find A, B and C.
Answers
the coordinates of A, B, and C on the line y = x - 6 are A = (6, 0), B = (1, -5), and C = (-7, -13).
How to find what is coordinate?
To find the coordinates of A, B, and C on the line y = x - 6, we need to substitute the given values of x in the equation and solve for y.
When x = 6, y = x - 6 = 6 - 6 = 0, so A = (6, 0).
When x = 1, y = x - 6 = 1 - 6 = -5, so B = (1, -5).
When x = -7, y = x - 6 = -7 - 6 = -13, so C = (-7, -13).
Therefore, the coordinates of A, B, and C on the line y = x - 6 are A = (6, 0), B = (1, -5), and C = (-7, -13).
A coordinate is a set of values that indicate the position of a point or object in space. In mathematics, coordinates are typically represented by ordered pairs or triplets of numbers, which represent the point's location on a plane or in space, respectively.
Coordinates can be used to describe the position of objects in a variety of fields, including mathematics, physics, and geography. For example, latitude and longitude coordinates are used to locate places on Earth's surface, and Cartesian coordinates are used in geometry to describe points on a plane.
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Related Questions
a random sample of 120 college seniors found that 30% of them had been offered jobs. what is the standard error of the sample proportion?
Answers
A random sample of 120 college seniors found that 30% of them had been offered jobs. the standard error of the sample proportion is 0.045.
When dealing with random samples and proportions, the standard error can be calculated using this formula:
SE_p =[tex]sqrt [ p × ( 1 - p ) / n ][/tex]
Where,
SE_p is the standard error of the sample proportion,
p is the sample proportion,n is the sample size.
Using the given information, a random sample of 120 college seniors found that 30% of them had been offered jobs. Hence, p=0.30 and n=120.
Substituting p=0.30 and n=120 in the above formula, we get:
SE_p = [tex]sqrt [0.30 × (1 - 0.30) / 120][/tex]
Simplifying,
SE_p = [tex]sqrt [0.21 / 120]SEp = 0.045[/tex]
Hence, the standard error of the sample proportion is 0.045.
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what is the value of x?
Answers
Applying the triangle proportionality theorem, the value of x is calculated as: D. 15.
How to Apply the Triangle Proportionality Theorem?The Triangle Proportionality Theorem (also known as the Side-Splitter Theorem) states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally
Based on the triangle proportionality theorem, we have:
x - 6 / 3 = x / 5
Cross multiply:
5(x - 6) = 3x
5x - 30 = 3x
Combine like terms to find the value of x:
5x - 3x = 30
2x = 30
x = 30/2
x = 15
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Let Y have a distribution function given by 0, Fly) = y <0 y 20. - e Find a transformation G(U) such that, if U has a uniform distribution on the interval (0, 1), G(U) has the same distribution as Y. G(U) = |(-In(1–u))(7
Answers
Let Y have a distribution function given by 0, Fly) = y <0 y 20.
To find a transformation G(U) such that if U has a uniform distribution on the interval (0, 1), G(U) has the same distribution as Y, follow these steps:
1. Identify the distribution function of Y:
F(y) = 0 for y < 0, and F(y) = 1 - e^(-y) for y ≥ 0.
2. Set F(y) equal to the cumulative distribution function (CDF) of a uniform distribution on the interval (0, 1):
F(y) = U.
3. Solve for y in terms of U to obtain the transformation G(U).
So, for y ≥ 0,
1 - e^(-y) = U
Now, solve for y:
e^(-y) = 1 - U
-y = ln(1 - U)
y = -ln(1 - U)
Therefore, the transformation G(U) is given by G(U) =. ln(1 - U)
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I need help quick! Quickest (correct) response gets brainliest!
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According to the question, it’s given that the sum of two numbers is 42.
On substituting the values -
[tex]\:\:\:\:\:\:\:\star \small \underline{ \boxed{ \sf{ x+\bigg(x-12\bigg) = 42}}}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf { x + x -12 = 42}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {2x -12 =42}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {2x = 42+12}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {2x = 54}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {x = \dfrac{54}{2}}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \boxed{ \tt{ \pmb{ \pink{x = 27}}}}\\[/tex]
Henceforth,the larger number is 27 and the smaller number is (x-12)=(27-12)=15.
[tex]\:\:\:\:\:\:\longrightarrow\underline{\rm{\sf Larger \: Number =\pink{ \underline{27}}}}[/tex]
[tex]\:\:\:\:\:\:\longrightarrow\underline{\rm{\sf Smaller \: Number =\pink{ \underline{15}}}}[/tex]
How do I work this out ?
Answers
Answer:
Step-by-step explanation:
First, I believe you would go in and mulitiply that 3 and the 1/6 and from there we will get 0.5. Next you are going to add the 2/5 to that 0.5 and you will get 0.9.
Answer: 0.9
this answer to the problem. Kind of tricky.
Answers
suppose we should solve the following equation:
[tex]s = \frac{13}{2} (12 + 75)[/tex]
which equals 565.5
the management of f xyzfinity cable estimates that the number of new subscribers (in thousands) next year in charlottesville is described by the random variable x and the number of new subscribers (in thousands) in albemarle county outside of charlottesville is described by the random variable y. if the joint probability density function is f(x,y)
Answers
To estimate the number of new subscribers in Charlottesville and Albemarle County, we'll use the given joint probability density function (pdf) f(x,y).
Here's a step-by-step explanation:
1. Identify the random variables: In this case, X represents the number of new subscribers (in thousands) in Charlottesville, and Y represents the number of new subscribers (in thousands) in Albemarle County outside of Charlottesville.
2. Understand the joint probability density function (pdf): The function f(x,y) describes the probability of having a specific number of new subscribers in both areas. The joint pdf is used to find the probability of a particular combination of X and Y values.
3. Determine the desired outcome: In this case, you want to estimate the number of new subscribers in both Charlottesville (X) and Albemarle County (Y) next year.
4. Calculate the probabilities: To find the probability of a specific combination of X and Y values, you'll need to evaluate the joint pdf f(x,y) at those values.
5. Analyze the results: Based on the probabilities you've calculated using the joint pdf, you can make informed estimates about the number of new subscribers in both areas next year.
By following these steps and using the joint probability density function f(x,y), you can estimate the number of new subscribers for F XYZ finity Cable in Charlottesville and Albemarle County next year.
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A line passes through the point (-8, 7) and has a slope of -5/4
Write an equation in slope-intercept form for this line.
Answers
The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept.
We are given that the line passes through the point (-8, 7) and has a slope of -5/4. So we can substitute these values into the slope-intercept form and solve for b:
y = mx + b
7 = (-5/4)(-8) + b
7 = 10 + b
b = -3
Therefore, the equation of the line in slope-intercept form is:
y = (-5/4)x - 3
ten percent of computer parts produced by a certain supplier are defective. what is the probability that a sample of 10 parts contains more than 3 defective ones?
Answers
The probability of a sample of 10 parts containing more than 3 defective ones is approximately 0.026.
We can use the binomial distribution to calculate the probability of getting more than 3 defective parts in a sample of 10 parts. Let X be the number of defective parts in the sample. Then X follows a binomial distribution with parameters n=10 and p=0.1, where n is the sample size and p is the probability of a part being defective.
We can calculate the probability of getting more than 3 defective parts is:
P(X > 3) = 1 - P(X ≤ 3) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3)]
Next, we can find that:
P(X > 3) = 0.026
Therefore, the probability of a sample of 10 parts containing more than 3 defective ones is approximately 0.026.
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Please Do it step by step without any explanation.
Answers
I need a bit of help
Answers
Substitute r = 2 and s = 3 into the expression:
5r + 7s - 3(r + s) + 4 = 5(2) + 7(3) - 3(2 + 3) + 4
= 10 + 21 - 15 + 4
= 20
Therefore, the value of the expression when r = 2 and s = 3 is 20.
Given:-
[tex] \texttt{r = 2}[/tex][tex] \: [/tex]
[tex] \texttt{s = 3}[/tex][tex] \: [/tex]
Solution:-
[tex] \texttt{5r+ 7s - 3( 2 + 3 ) + 4}[/tex][tex] \: [/tex]
put the given values in the equation
[tex] \texttt{5( 2 ) + 7 ( 3 ) - 3( 2 + 3 ) + 4.}[/tex][tex] \: [/tex]
[tex] \texttt{10 + 21 - 6 - 9 + 4}[/tex][tex] \: [/tex]
[tex] \texttt{31 - 15 + 4}[/tex][tex] \: [/tex]
[tex] \texttt{16 + 4}[/tex][tex] \: [/tex]
[tex] \texttt{ \red{20}} \: [/tex][tex] \: [/tex]
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
hope it helps ⸙
a line of length l is scaled of 1.5 to produce a segment with length m. the new segment is then scaled by a factor of 1.5 to give a segment of length n.what scale factor takes the segment of length l to the segment of length n
Answers
The scale factor that takes the segment of length l to the segment of length n is 2.25.
What is length?Length is a measurement of the distance between two points, usually expressed in units such as inches, feet, yards, meters, and kilometers. It is used to measure the size of objects, the distance between objects, and the length of a path or route. Length is one of the four fundamental units in the metric system, along with mass, time, and temperature.
This can be determined by a process of successive scaling. The initial segment of length l is first scaled by a factor of 1.5 to give a segment of length m. This segment is then scaled by a factor of 1.5 to give a segment of length n. Multiplying the two scaling factors together, 1.5 x 1.5, gives 2.25, which is the scale factor from the initial segment of length l to the final segment of length n.
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The scale factor that takes the segment of length l to the segment of length n is 2.25.
What is length?Length is a measurement of the distance between two points, usually expressed in units such as inches, feet, yards, meters, and kilometers. It is used to measure the size of objects, the distance between objects, and the length of a path or route. Length is one of the four fundamental units in the metric system, along with mass, time, and temperature.
This can be determined by a process of successive scaling. The initial segment of length l is first scaled by a factor of 1.5 to give a segment of length m. This segment is then scaled by a factor of 1.5 to give a segment of length n. Multiplying the two scaling factors together, 1.5 x 1.5, gives 2.25, which is the scale factor from the initial segment of length l to the final segment of length n.
To calculate the scale factor that takes the segment of length l to the segment of length n, the following formula can be used:
n = (1.5)²×l
Therefore, the scale factor is (1.5)² = 2.25. This means that the segment of length l needs to be scaled by a factor of 2.25 in order to produce the segment of length n.
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^-AB IS THE MID SEGMENT of triangle CDE what is the value of x? Show all work please
Answers
In response to the stated question, we may state that As a result, the trigonometry value of x is 11/4.
what is trigonometry?The study of the connection between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their possible applications in calculations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle characteristics, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric forms.
Because AB is the midpoint of triangle CDE, it is parallel to CD and equal to half of CD.
As a result, we have:
AB = 1/2 CD
We also know that AD has a length of 3x - 1 and DB has a length of x + 4.
Given that AB is the mid-segment, we may write:
1/2 (AD + DB) = AB
In place of the values we have:
3x - 1 + x + 4 = 2AB
4x + 3 = 2AB
We also know that AB = 7, therefore we may use this number instead:
4x + 3 = 2(7) (7)
4x + 3 = 14
4x = 11
x = 11/4
As a result, the value of x is 11/4.
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Sara cut a 2 1/2 meter rope to hang a swing for her sister. How many centimeters is the rope
Answers
Length of the Sara's rope of 2 1/2 meter in centimeter is 25 centimeters.
To convert 2 1/2 meters to centimeters, we can use the conversion factor 1 meter = 100 centimeters. This means that:
2.5 meters = 2.5 x 100 centimeters
= 250 centimeters
Therefore, the length of the rope is 250 centimeters. It's important to understand and be able to convert between different units of measurement, as this is a common task in many fields such as science, engineering, and finance. For example, in science, it's important to be able to convert between different units of length, mass, and volume when making measurements or analyzing data. Similarly, in finance, it's common to convert between different currencies or units of time when dealing with investments or loans. Being able to make these conversions accurately is essential to avoid errors or misunderstandings. In this case, converting the length of the rope from meters to centimeters allows us to work with a more convenient unit for the task at hand, which is hanging a swing for Sara's sister.
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if you have to choose a complete meal from one of 3 meats, one of 4 vegetables, and one of 7 desserts, how many different choices are there? question 17 options:
Answers
3 meats x 4 vegetables x 7 desserts = 84 choices.
There are 84 different choices for a complete meal from the given options. To calculate this, use the formula: number of choices
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Terrance and his three friends earned $359 in August, $522 in July, and $420 in September selling lemonade. How much would they each earn if they divided their earnings equally?
Answers
In Linear equation, 260.2 would they each earn if they divided their earnings equally.
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
Equations with variables of power 1 are referred to be linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.
three friends earned $359 in August, $522 in July, and $420 in September
= $359 + $522 $ 420
= 1301/5 = 260.2
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the atmospheric carbon dioxide levels in parts per million (ppm) in a town can be modeled using the function defined by where is in years and corresponds to 1950. find and interpret the result. round to 2 decimal places as needed. answer: with unit
Answers
The atmospheric carbon dioxide levels in parts per million (ppm) in a town can be modeled using the function defined by , where is in years and corresponds to 1950.
Substituting in gives , which means that the atmospheric carbon dioxide levels in parts per million (ppm) in the town is 386.50 in 2019.
This means that since 1950, the atmospheric carbon dioxide levels in parts per million (ppm) in the town have increased by 386.50.
This is a significant increase and reflects the growing levels of atmospheric carbon dioxide emissions globally due to human activity, leading to climate change.
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HELP HELP HELP
30 points
Answers
A sketch of the graph of the function for a maximum of 4000 calculators is shown below.
The value of C(30) is equal to $12,710.
It cost $12,710 to produce 30 calculators.
C(0) is $11,750, which means that the fixed cost before any calculator is produced is $11,750.
The number of calculators that can be produced for $31,270 is 610 calculators.
The amount of profit that would be made if all the calculators are sold is $295,550.
How to sketch the graph of the function?In order to sketch the graph of the function on a coordinate plane, we would use an online graphing calculator to plot the given linear equation C(n) = 11,750 + 32n with a maximum of 4000 calculators.
In this scenario and exercise, the cost in dollars would be plotted on the y-axis of the graph while the number of calculators would be plotted on the x-axis of the graph.
For the value of C(30), we have:
C(n) = 11,750 + 32n
C(30) = 11,750 + 32(30)
C(30) = 11,750 + 960
C(30) = $12,710.
For the value of C(0), we have:
C(n) = 11,750 + 32n
C(0) = 11,750 + 32(0)
C(0) = 11,750 + 0
C(0) = $11,750.
This ultimately implies that, the fixed cost (y-intercept) is at (0, 11,750) and it means that the fixed cost before any calculator is produced is $11,750.
When cost is $31,270, the number of calculators can be determined as follows;
31,270 = 11,750 + 32n
32n = 31,270 - 11,750
32n = 19,520
n = 19,520/32
n = 610 calculators.
Lastly, we would determine the amount of profit as follows;
Profit = Selling price - Cost price
Profit = 2600(165) - [11,750 + 32(165) + 14,000 + 24500]
Profit = 429,000 - 133,450
Profit = $295,550.
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Solve the system of equations.
–6x + y = –21
2x − 1
3
y = 7
What is the solution to the system of equations?
(3, 3)
(2, –9)
infinitely many solutions
no solutions
Answers
The closest option is (A) (3,3), which is the correct solution to the system of equations.
EquationsTo find the solution to the system of equations, we need to substitute the value of y in the first equation with the value given in the second equation:
-6x + y = -21 ...(1)
2x - 1/3 y = 7 ...(2)
Substituting y=7 in the first equation, we get:
-6x + 7 = -21
Simplifying the above equation:
-6x = -28
Dividing both sides by -6, we get:
x = 28/6 = 14/3
Substituting x=14/3 and y=7 in the second equation, we get:
2(14/3) - 1/3(7) = 7
Simplifying the above equation, we get:
28/3 - 7/3 = 7
21/3 = 7
Therefore, the solution to the system of equations is (14/3, 7).
Hence, the answer is not in the given options, but the closest option is (A) (3,3), which is not the correct solution to the system of equations.
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solve the inequality
`-5x+2<=12`
Answers
The solution for the inequality is obtained as: x =>-2
Explain about the inequality of the number?In mathematics, "inequality" refers to a relationship between equations or values that is not equal to each other. Hence, inequality emerges from a lack of balance.
We can see the values that an inequality represents by placing it on a number line. On a number line, inequalities are represented by tracing a straight path and designating the end points by using an open or closed circle.The given inequality
-5x + 2 <= 12`
Subtract each side by - 2
-5x + 2 - 2 <= 12 - 2
-5x <= 10
Dive each side by -5 (it will reverse the sign)
-5x / -5 => 10/-5
x => -2
Thus, the solution for the inequality is obtained as: x =>-2
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an individual was making blankets for some friends. the individual used 7.2 feet of fabric to make one blanket. how many feet of fabric will the individual need to make 11 blankets (round to the nearest whole number)?
Answers
The number of feet of fabric will the individual need to make 11 blankets is 79.2 feet
To find out how many feet of fabric the individual will need to make 11 blankets, we can use the unitary method.
We know that the individual used 7.2 feet of fabric to make one blanket. So, to find out how much fabric is needed to make 11 blankets, we can set up a proportion
7.2 feet of fabric is needed for 1 blanket
x feet of fabric is needed for 11 blankets
We can solve for x by cross-multiplying and then dividing
7.2 × 11 = x
x = 79.2 feet
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you are planning a day at the beach and want to know if the water will be pleasant or not. suppose there is a 0.51 chance of the water being cold and a 0.57 chance of the waves being turbulent. suppose also that there is a 0.39 chance that the water is cold given that the waves are turbulent. what is the probability that the waves are turbulent given that the water is cold? round your answer to the nearest millionth, if necessary.
Answers
The probability that the waves are turbulent given that the water is cold is approximately 0.4353.
Suppose the water has a 0.51 chance of being cold and the waves have a 0.57 chance of being turbulent. In the same way, suppose that given the waves are turbulent, the water has a 0.39 chance of being cold. What is the chance that the waves are turbulent given that the water is cold?
To answer the question, we must use Bayes' theorem :P (A|B) = P (B|A) × P (A) / P (B)
Where A and B are arbitrary events. P (A) and P (B) are the probabilities of events A and B happening separately.
P (B|A) is the probability of event B happening, given that event A has happened. P (A|B) is the probability of event A happening, given that event B has happened.
We must find P (waves are turbulent | water is cold), which is P (B|A). We know that:
P (water is cold) = 0.51
P(waves are turbulent) = 0.57
P(water is cold | waves are turbulent) = 0.39
Using Bayes' theorem, we can calculate P (waves are turbulent | water is cold) as:
P (waves are turbulent | water is cold) = P (water is cold | waves are turbulent) × P (waves are turbulent) / P (water is cold)
P(waves are turbulent | water is cold) = 0.39 × 0.57 / 0.51
P (waves are turbulent | water is cold) = 0.4353 (rounded to the nearest millionth)
Therefore, the probability that the waves are turbulent given that the water is cold is approximately 0.4353.
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2x-6=2x+6 solve for x
Answers
Answer:
no solution
Step-by-step explanation:
2x - 6 = 2x + 6 ( add 6 to both sides )
2x = 2x + 12 ( subtract 2x from both sides )
0 = 12 ← false statement
this false statement indicates the equation has no solution
Help me with the even numbers. 2,4,6,and8
Answers
2= 2
4=10
i cant see 6 or 8
The senior class put on a talent show as a fundraiser. Tickets cost $7 for adults and $4 for students. 185 people attended the show and the ticket sales totaled $1,115. How many adults attended the show? How many students attended the show?
Answers
9. The linear regression equation is = 34.38x - 91.75. Use the equation to predict how far this
4.38x-91-75 Use
person will travel after 10 hours of driving.
Answers
The answer of the given question based on the linear regression is , the predicted distance the person will travel after 10 hours of driving is approximately 252.05 miles.
What is Distance?Distance is measurement of length between the two points or objects. It is a scalar quantity that only has a magnitude and no direction. In mathematics, distance can be measured in various units such as meters, kilometers, miles, or feet, depending on the context.
Distance can be calculated using the distance formula, which is based on the Pythagorean theorem in two or three dimensions.
Assuming the equation you meant to write is y = 34.38x - 91.75, where y is the predicted distance traveled in miles and x is the number of hours driven, we can use this equation to predict how far the person will travel after 10 hours of driving:
y = 34.38x - 91.75
y = 34.38(10) - 91.75
y = 343.8 - 91.75
y = 252.05
Therefore, the predicted distance the person will travel after 10 hours of driving is approximately 252.05 miles.
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During 10 hours of driving, the projected distance according to linear regression is roughly 252.05 miles.
What is Distance?The term "distance" refers to the length between two points or objects. Having merely a magnitude and no direction, it is a scalar quantity. Depending on the situation, distance in mathematics can be expressed in a variety of ways, including meters, kilometers, miles, or feet.
The distance formula, which depends on the Pythagorean theorem in either two or three dimensions, can be used to compute distance.We may use this equation to forecast how far the individual would go after 10 hours of driving, assuming the equation you meant to write is
y = 34.38x - 91.75, where y is the expected distance travelled in miles and x is the number of hours driven:
y = 34.38x - 91.75
y = 34.38(10) - 91.75
y = 343.8 - 91.75
y = 252.05
The estimated distance that the driver will cover after 10 hours on the road is 252.05 miles.
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The complete question is,
The equation for linear regression is = 34.38x - 91.75. Calculate this person's estimated distance after 10 hours of driving using the equation: 4.38x-91-75.
I need help finding the subsets and proper subsets. Please help it’s due tonight!!
Answers
Answer:
Step-by-step explanation:
# elements = 292 - 268 - 1 = 23 elements
G has 2^23 subsets = 8388608
G has 2^23 - 1 proper subsets = 8388607
What is the area of the parallelogram
I need an answer asap!!!!!!!
Answers
Answer: A = bh
Step-by-step explanation
Area of a parallelogram is base multiplied by the height .
true/false: let a be an n \times n matrix. a is diagonalizable if it has n eigenvalues, counting multiplicities.
Answers
The given statement " let a be an n \times n matrix. a is diagonalizable if it has n eigenvalues, counting multiplicities." is False. Because Matrix has only one linearly independent eigenvector.
Having n eigenvalues, counting multiplicities, only guarantees that A is diagonalizable over the field of complex numbers, but not necessarily over other fields such as the real numbers. The correct statement is Let A be an n × n matrix. A is diagonalizable if and only if it has n linearly independent eigenvectors.
For example, the matrix:
A = [0 1; -1 0] hhggjjjhjhjhj
has eigenvalues i and -i, but it is not diagonalizable over the field of real numbers, since it has only one eigenvector which is also linearly independent. However, it is diagonalizable over the field of complex numbers. So, the given statement is false.
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An arithmetic series of A has first term a and common difference d.
The sum of Sn of the first n termof A is given by Sn=(15+2n)
(a) Find the value of a and d
(b) Find the 20th term of A
Given that S2p - 2Sp = 1 + S(p-1)
(c) find the value of p
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Answers
Answer:
(a) To find the value of a and d, we use the formula for the sum of first n terms of the arithmetic series A which is given by:Sn = n/2[2a + (n-1)d]We are also given that Sn = 15 + 2n. So we can equate these two expressions to get:15 + 2n = n/2[2a + (n-1)d]Multiplying both sides by 2 and simplifying, we get:30 + 4n = n[2a + (n-1)d]Expanding the brackets and simplifying, we get:2an + nd - d + 30 = 2n^2Rearranging terms, we get:2a = 2n^2 - nd + d - 30Now we also know that the first term of the series A is a. So we can substitute this value of a in the formula above to get:a = (2n^2 - nd + d - 30)/2Simplifying, we get:a = n^2 - (n-1)d - 15Therefore, we have found the values of a and d in terms of n. (b) To find the 20th term of A, we use the formula for the nth term of an arithmetic series which is given by:an = a + (n-1)dSubstituting the value of a and d that we found in part (a) we get:a20 = (20^2 - 19d - 15) + 19dSimplifying, we get:a20 = 391 - dTherefore, the 20th term of A is given by a20 = 391 - d.(c) Given that S2p - 2Sp = 1 + S(p-1), we can use the formula for the sum of first n terms of an arithmetic series which we used in part (a) to get:2p/2[2a + (2p-1)d] - 2p/2[2a + (p-1)d] = 1 + p/2[2a + (p-2)d]Simplifying, we get:2apd = d(p^2 - 3p + 2)Dividing both sides by d and simplifying, we get:2ap = p^2 - 3p + 2Rearranging terms, we get:p^2 - 3p + (2-2ap) = 0This is a quadratic equation with coefficients a=1, b=-3, and c=2-2ap. We can use the quadratic formula to solve for p:p = [3 ± sqrt(9 - 4(1)(2-2ap))]/2Simplifying, we get:p = [3 ± sqrt(4ap + 1)]/2Therefore, we have found the value of p in terms of a.