The standard form of the equation of the circle with the given characteristics is (x - 6)² + (y - 1)² = 65.
What is an equation?
An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.
Given, center: (6, 1) and solution Point: (5,9)
By solution point, it means that the circle passes through that point. Hence, the radius of the circle is the distance between (6, 1) and (5,9)
Distance formula is given by
D = √[(x2 - x1)² + (y2 - y1)²]
D= √[(5 - 6)² + (9 - 1)²] = 8.062
r= 8.062 or r²=65
We have, equation of the circle centered ant (h, k) and radius ‘r’ as
(x - h)² + (y - k)² = r².
=> (x - 6)² + (y - 1)² = 65.
The standard form of the equation of the circle with the given characteristics is (x - 6)² + (y - 1)² = 65.
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from the top of a 250-ft lighthouse, the angle of depression to a ship in the ocean is 21°. how far is the ship from the base of the lighthouse? (round your answer to the nearest foot.)
The ship is at a distance of 651 feet away from the lighthouse.
Therefore the answer is 651.
Let's call the distance from the base of the lighthouse to the ship "d". Then, using basic trigonometry:
tan∅ = opposite side/ adjacent side
So here opposite side is the height of the lighthouse and adjacent side is the distance from the base of the lighthouse.
tan(21°) = height of lighthouse/ d
d = height of lighthouse/ tan(21°)
d = 250 / tan(21°)
d = 250 / 0.384 = 651.3 ft
Rounding to the nearest foot, the ship is 651 ft away from the base of the lighthouse.
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ASAP. Need the question soon—25 points given.
Answer:
s=8
Step-by-step explanation:
The two angles are vertical angles and vertical angles are equal.
110 = 8s-2
Add 2 to each side.
110+2 = 8s-2+2
112 = 8s
Divide each side by 8
112/8=8s/8
14=s
80 percent of students finished their assignment on time. if 30 turned theirs in late, how many total students were in the class?
Answer:
150
Step-by-step explanation:
80% on time so 20% late, 30 late
30/20% = 150
How heavy is 150 pounds? How much does 150 pounds weigh in kilograms? 150 lb to kg conversion.
The conversion of 150 pounds weight in kilograms 68.039.
What is meant by weight?
The gravitational force that pulls a body toward the earth or another celestial body; it is equal to the mass multiplied by the local gravitational acceleration.Weight, which is expressed in newtons, is a unit of measurement for the gravitational force acting on an object. The weight of a bird with a mass of 15 g changes with the strength of the gravitational force acting on it, and would differ significantly if it were measured, for instance, on the Moon rather than on Earth.On Earth compared to the Moon, an object would weigh differently as a result of gravity. The following is an explanation of the weight formula: Weight is calculated as mass multiplied by gravity. W = mg is the formula for this.To learn more about weight refer to
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9) Refer to the table which summarizes the results of testing for a certain disease.
Positive Test Negative Test
Result Result
Subject has the disease
Subject does not have the disease
111
11
4
172
one of the results is randomly selected, what is the probability that it is a false negative (test
indicates the person does not have the disease when in fact they do)? What does this probability
suggest about the accuracy of the test?
A) 0. 591, The probability of this error is high so the test is not very accurate.
B) 0. 0348, The probability of this error is low so the test is fairly accurate.
C) 0. 0134; The probability of this error is low so the test is fairly accurate.
D) 0. 0369; The probability of this error is low so the test is fairly accurate.
The probability that it is a false negative (test indicates the person does not have the disease when in fact they do) is 0.0348.
This probability indicates that the test's accuracy is 0. 0348; as the likelihood of this inaccuracy is minimal, the test is quite accurate.
The likelihood that a randomly chosen result will be a false negative, meaning that the test will show that the person does not have the condition while in fact they do:
[tex]P(NP|D) = \frac{4}{115} \\\\P(NP|D) = 0.0348[/tex]
We may infer that the occurrence is unlikely to occur and that the test is reasonably accurate because the probability is less than 0.0500.
Probability is simply the possibility that something will happen. When we don't know how something will turn out, we might talk about the possibility of one result or the likelihood of several. The study of occurrences that fit into a probability distribution is known as statistics.
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Need answers before Friday
The discounted prices are given below.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
Example:
50% = 50/100 = 1/2
25% = 25/100 = 1/4
20% = 20/100 = 1/5
10% = 10/100 = 1/10
We have,
a)
Price = $1.20
Off = 25% = 1/4
Discounted amount = 1/4 x 1.20 = $0.3
Discounted price = 1.20 - 0.3 = $0.9
b)
Price = $18
Off = 25% = 1/4
Discounted amount = 1/4 x 18 = $4.5
Discounted price = 18 - 4.5 = $13.5
c)
Price = $150
Off = 30% = 0.3
Discounted amount = 0.3 x 150 = $45
Discounted price = 150 - 45 = $105
d)
Price = $20
Off = 40% = 0.4
Discounted amount = 0.4 x 20 = $8
Discounted price = 20 - 8 = $12
e)
Price = $2.20
Off = 10% = 1/10
Discounted amount = 1/10 x 2.20 = $0.22
Discounted price = 2.20 - 0.22 = $1.98
f)
Price = $1.30
Off = 50% = 1/2
Discounted amount = 1/2 x 1.30 = $0.65
Discounted price = 1.30 - 0.65 = $0.65
Thus,
The discounted prices are given above.
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93 x 10 with the power of 3
Answer:93000
—————————————-
Someone please help me on thisss
82.1% of cincinnati households have a computer. if 111,804 households in cincinnati have a computer, how many households are there altogether in cincinnati? show your work. round to the nearest whole number.
82.1% of cincinnati household have a computer . If 111,804 households in cincinnati have a computer then 136 households are there altogether in cincinnati.
Given that, 82.1% of Cincinnati households have a computer.
111,804 households in Cincinnati have a computer.
Ratios: It is the realtion between two numbers which shows how much bigger on quantity is than another.
Proportion: It is the an equality of two ratios.
Let total number of houses be x;
According to ratio and proportions we have,
(111,804/x)*100=percentage of Cincinnati households having a computer.
then (111,804/x)*100=82.1
cross multiplying we have
x=(111,804*100)/82.1
x=136 households (Round to the nearest whole number.)
so total households present are 136
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82.1% of Cincinnati households have a computer. If 111,804 households in Cincinnati have a computer then 136 households are there altogether in Cincinnati.
Given that, 82.1% of Cincinnati households have a computer.
111,804 households in Cincinnati have a computer.
Ratios: It is the relation between two numbers that shows how much bigger one quantity is than another.
Proportion: It is the equality of two ratios.
Let the total number of houses be x;
According to the ratio and proportions we have,
(111,804/x)*100=percentage of Cincinnati households having a computer.
then (111,804/x)*100=82.1
cross multiplying we have
x=(111,804*100)/82.1
x=136 households (Round to the nearest whole number.)
so total number of households present are 136
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PLEASE HELP! What is another way to write the expression x(14 + 19) + 30
Answer:
33x+30
Step-by-step explanation:
If you distribute the x to 14 and 19, you will end up with:
14x+19x+30
Then, if you add the two x values together, you'll end up with:
33x+30
Another way you can do it is adding the numbers in the parentheses first:
x(33)+30
Then distribute the x:
33x+30
The value of the digit 2 in the number 25.381 is 10 times greater then the value of the digit 2 in which of the following numbers
The value of the digit 2 in the number 25.381 is 10 times greater than the value of the digit 2 in 52.
The correct option is A.
What is a place value?Place value, which is related to a digit's location in a number, is the amount that each digit is worth.
Given:
We have a decimal number 25.381.
From the given choices;
We have number 52.
In the number 52:
The place value of 2 is 2.
The value of the digit 2 in the number 25.381 is 10 times greater than the value of the digit 2 in 52.
Therefore, the number is 52.
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The complete question:
The value of the digit 2 in the number 25.381 is 10 times greater than the value of the digit 2 in which of the following numbers:
A) 52.
B) 25534.675
La expresión algebraica de la relación sigue siendo y = 10x, donde x es la cantidad
de entregas a domicilio que realizan, mientras y es el ingreso en soles que
obtienen por ellas.
Representa, mediante un gráfico de valores, la relación entre la cantidad de
envíos a domicilio y el ingreso (S/).
¿Consideras que, para representar la función mediante un gráfico, primero e
fundamental dibujar el plano cartesiano? Justifica tu respuesta.
Algunos estudiantes dibujan la siguiente gráfica, asignan valores al eje de la
abscisas de uno en uno y al eje de las ordenadas de 10 en 10.
Sí, considero que para representar la función mediante un gráfico, es necesario primero dibujar el plano cartesiano. Esto es porque el plano cartesiano permite representar gráficamente una relación entre dos variables.
Esto también le permite al usuario visualizar los valores que están relacionados y ver cómo se comportan las variables en relación con el otro. El plano cartesiano también le permite al usuario ver los valores que se encuentran fuera del rango de los datos y ayuda a entender mejor la relación entre los datos.
El hecho de asignar valores al eje de la abscisas de uno en uno y al eje de las ordenadas de 10 en 10 no es necesario para representar la función mediante un gráfico, ya que esto depende de las escalas que se elijan para los ejes. Por ejemplo, si la cantidad de envíos a domicilio es el eje x y el ingreso es el eje y, el usuario puede elegir diferentes escalas para los ejes para representar la relación. Por ejemplo, el usuario puede elegir una escala de 1 en 1 para el eje x y una escala de 10 en 10 para el eje y, o una escala de 2 en 2 para el eje x y una escala de 5 en 5 para el eje y.
English version:Yes, I believe that in order to represent the function using a graph, it is essential to first draw the Cartesian plane. This is because the Cartesian plane allows for graphically representing a relationship between two variables. This also allows the user to visualize the values that are related and see how the variables behave in relation to one another. The Cartesian plane also allows the user to see the values that are outside of the range of the data and helps to better understand the relationship between the data.
Assigning values to the x-axis one by one and to the y-axis 10 by 10 is not necessary to represent the function using a graph, as this depends on the scales chosen for the axes. For example, if the number of deliveries is the x-axis and the income is the y-axis, the user can choose different scales for the axes to represent the relationship. For example, the user can choose a scale of 1 to 1 for the x-axis and a scale of 10 to 10 for the y-axis, or a scale of 2 to 2 for the x-axis and a scale of 5 to 5 for the y-axis.
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The diameter of a cone's circular base measures 10 inches, and the slant height of the cone is 8 inches.
What is the approximate surface area of the cone?
Responses
146.3 in²
146.3 in²
204.1 in²
204.1 in²
314.2 in²
314.2 in²
565.2 in²
The surface area of the cone with a slant height of 8in and diameter of 10in is 204.1 in².
What is the surface area of the cone?A cone is simply a 3-dimensional geometric shape with a flat base and a curved surface pointed towards the top.
The surface area of a cone is expressed as;
A = πrl + πr²
Given that;
Diameter d = 10inRadius = diameter/2 = 10/2 = 5inSlant height h = 8inSurface area A = ?Plug the given values into the above formula and simplify
A = πrl + πr²
A = ( 3.14 × 5 × 8 ) + ( 3.14 × 5² )
A = 125.6 + 78.5
A = 204.1 in²
Therefore, the surface area is 204.1 in²
Option B) 204.1 in² is the correct answer.
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Perform the following operation
and express the answer in
scientific notation.
The solution of the expression in scientific notation is 1.5 * 10^-2
How to evaluate the expressionFrom the question, we have the following parameters that can be used in our computation:
6.0 * 10^4 / 4.0 * 10^8
Divide 6 by 4
So, we have
6.0 * 10^4 / 4.0 * 10^8 = 1.5 * 10^4/10^8
Evaluate the final quotient
6.0 * 10^4 / 4.0 * 10^8 = 1.5 * 10^-2
Hence, the solution is 1.5 * 10^-2
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find the following limit or state that it does not exist. x-9
The limit of the function [tex]\lim_{x \to9}[/tex] [( x² - 81 ) / ( x - 9 ) ] is equal to 18.
[tex]\lim_{x \to9}[/tex] [( x² - 81 ) / ( x - 9 ) ]
Substitute the limit x tends to 9 in the function we have,
= ( 9² - 81 ) / ( 9 - 9 )
= ( 81 - 81 ) / 0
= 0 /0
0 /0 form is not defined term.
Simplify the function before substituting the limit we have,
[tex]\lim_{x \to9}[/tex] [( x² - 81 ) / ( x - 9 ) ]
= [tex]\lim_{x \to9}[/tex] [ ( x ² - 9² ) / ( x - 9 ) ]
= [tex]\lim_{x \to9}[/tex] [ ( x - 9 ) ( x + 9 ) / ( x - 9 ) ]
= [tex]\lim_{x \to9}[/tex] ( x + 9 )
= 9 + 9
= 18
Therefore, the value of the limit for the function [tex]\lim_{x \to9}[/tex] [( x² - 81 ) /( x - 9) ] is equal to 18.
The given question is incomplete, I answer the question in general according to my knowledge:
Find the following limit or state that it does not exist.
[tex]\lim_{x \to9} \frac{(x^{2} - 81)}{(x - 9)}[/tex].
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evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) 41 x3 − 27 dx
The integral of 41x^3 - 27 dx is 41/4 x^4 - 27x + c.
Calculating an integral is called integration. To determine the functions that will characterize the area, displacement, and volume that result from a combination of small data that cannot be measured separately, integrals are calculated.
The concept of limit is employed broadly in calculus whenever algebra and geometry are applied. Limits assist us in analyzing how points on a graph behave, such as how they approach one another until their distance approaches zero.
The indefinite integral of 41x^3 - 27 dx is given by:
∫(41x^3 - 27)dx = 41/4 x^4 - 27x + c
where c is the constant of integration.
Correct Question :
evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) 41x^3 - 27 dx
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Classify the following triangle. Check all that apply. Angle 120
In the given triangle the length of the two sides of the triangle is equal hence the triangle is an isosceles triangle, and the angle is 120 degree hence the triangle is obtuse triangle.
What is an isosceles triangle?An isosceles triangle is one with two equally sized sides. Also equal are the two angles that face the two equal sides. In other terms, an isosceles triangle is a triangle with two sides that are the same length.
If the sides AB and AC of a triangle ABC are equal, then the triangle ABC is an isosceles triangle with B = C. "If the two sides of a triangle are congruent, then the angle opposite to them is likewise congruent," states the theorem that characterizes the isosceles triangle.
In the given triangle the length of the two sides of the triangle is equal hence the triangle is an isosceles triangle, and the angle is 120 degree hence the triangle is obtuse triangle.
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A company has two large computers. The slower computer can send all the company's emails in 30 minutes. The faster computer can complete the same job in 15 minutes. If both computers are working together, how long will it take them to do the job?
The two computers simultaneously can send the email in 10 minutes.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the company has two large computers. The slower computer can send all the company's emails in 30 minutes. The faster computer can complete the same job in 15 minutes.
The time taken by both the computers together will be:-
1 / t = 1 / 30 + 1 / 15
1 / t = ( 1 + 2) / 30
1 / t = 3 / 30
1 / t = 1 / 10
t = 10 minutes
Hence, the time taken by both computers will be 10 minutes.
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The value of a certain car after t years is modeled by the expresslon 15,000(0. 7)^ What are the initial cost, and the rate of depreciation of this car?
The initial cost of the car was $15,000 and the rate of depreciation is 30%.
The equation [tex]15,000(0.7)^t[/tex] models the value of a car after t years. The value of the car after 0 years, or its initial cost, is [tex]15,000(0.7)^0[/tex] = 15,000. The rate of depreciation can be calculated as 1 - 0.7, which equals 0.3 or 30%. This means that each year, the value of the car decreases by 30% of its previous value. This model assumes a constant rate of depreciation, which may not always reflect real-world situations. However, it provides a simple estimate of the decline in value over time for this car.
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which of these statements do you think is true?
(a) Exactly one of these statements is false
(b) Exactly two of these statements are false
(c) Exactly three of these statements are false
(d) Exactly four of these statements are false
answer:
a. (a)
b. (b)
c. (c)
d. (d)
All the given options are paradoxical, and it's not possible to determine their truth value as they depend on their own veracity.
The statement "Exactly one of these statements is false" can be referred to as the liar paradox. It is a classic example of self-referential statements, where the truth of the statement depends on its own veracity.
In this paradox, if the statement is true, then it must be false, and if the statement is false, then it must be true. Hence, this statement creates a paradox and it's impossible to determine its truth value.
In the case of options (b), (c), and (d), it's also not possible to determine their truth value as they too depend on the truth value of the other statements, which are paradoxical.
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The upcoming championship high school football game is a big deal in your little town. The problem is, it is being played in the next biggest town, which is two hours away! To get as many people as you can to attend the game, you decide to come up with a ride-sharing app, but you want to be sure it will be used before you put all the time in to creating it. You determine that if more than three students share a ride, on average, you will create the app. You conduct simple random sampling of 20 students in a school with a population of 300 students to determine how many students are in each ride-share (carpool) on the way to school every day to get a good idea of who would use the app. The following data are collected:
6 5 5 5 3 2 3 6 2 2
5 4 3 3 4 2 5 3 4 5
The parameter used in the probability is the average number of students represented by u.
How to calculate the probability?The confidence interval based on the information will be:
= 3.85 - 2.09(1.348 / ✓20)
= 3.22
Also, 3.85 + 2.09(1.348 / ✓20) = 4.48
The confidence interval simply means that one is 95% confident that the true mean is between 3.22 and 4.48.
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in assessing the validity of a test, you randomly assign 100 individuals to two groups: one group assesses intelligence through a newly developed test, and a second group assesses intelligence through a traditional intelligence test that has been shown to be valid. you then conduct a statistical analysis to see if there are significant differences between the groups. your hypothesis states that there will be no significant differences. this type of validity is called:
This type of validity is called concurrent validity.
Concurrent validity refers to the extent to which a new test measures the same construct as an established, well-validated test. In this scenario, the newly developed test is being compared to a traditional intelligence test that has already been shown to be valid, and the hypothesis is that there will be no significant differences between the results obtained from the two tests. By comparing the scores of the two groups, the researchers are attempting to determine whether the new test is measuring the same construct (intelligence) as the established test.
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What advantage does the involvement of citizen scientists have in measuring biodiversity? Explain your answer.
Answer: citizen scientists are already providing large amounts of data for monitoring biodiversity, but they could do much more, according to a new study published in the journal Biological Conservation, which suggests that citizen science has the potential to contribute much more to regional and global assessments of biodiversity. Citizen scientists are regular people who provide data or input to science, for example by monitoring species in their community or examining satellite imagery for evidence of deforestation or land use change.
Step-by-step explanation:
A right square pyramid has a volume of 12 cubic centimeters. If the height remains fixed but the sides of the base are multiplied by 3, what is the volume of the new pyramid?
A. 36 cm
B. 48 cm
C. 108 cm
D. 72 cm
If the height of a square pyramid remains fixed while its sides of the base is multiplied by 3, then the volume of the new pyramid is 108 cm³ (option C)
The volume of a square pyramid is given by:
V = A . h = a² . h
Where;
A = the area of the base = a²
h = height of the pyramid
a = sides of the base
In this problem,
V = 12 cm³
a² . h = 12 .... (equation 1)
Now, the sides of the base are multiplied by 3. Hence, the new volume will be:
V = (3a)² . h
= 9 a² . h
from equation 1 we know that a² . h = 12. Hence,
V = 9 x 12 = 108 cm³
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Your school is selling tickets for the band concert. On the first day of ticket sales the school sold five adult tickets and three student tickets for a total of $18. The school took in $22 on the second day by selling seven adult tickets and one student ticket.
What is the price of a student ticket?
Adult ticket costs $3 and the student $1.
What is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.
here, we have,
1) Gathering the data
1st day
5a +3s =18
2nd day:
7a +1s= 22
2) Let's set a system of Linear equations, given those two equations:
solving we get,
a=3
s=1
3) Hence, the adult ticket costs $ 3 and the student $1.
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a. Find the ratio of the number of nonfiction book to the number of fiction books give your answer in fraction form. Give your answer in fraction form
b. How many times the number of nonfiction books is the number of fiction books? Give your answer in fraction form.
c. Suppose the number fiction books is 2/7 times the number of nonfiction books what would be the ratio of the number of nonfiction books to the number of TOTAL number of books give your answer in fraction form.
The ratio of nonfiction to fiction books is 49 : 2.
What are ratio and proportion?In its most basic form, a ratio is a comparison between two comparable quantities.
There are two types of proportions One is the direct proportion, whereby increasing one number by a constant k also increases the other quantity by the same constant k, and vice versa.
If one quantity is increased by a constant k, the other will decrease by the same constant k in the case of inverse proportion, and vice versa.
Given, The number of fiction books is 2/7 times the number of nonfiction books.
So, If we have 7 nonfiction books then we have 2/7 fiction books.
Therefore, The ratio Nonfiction : Fiction = 7 : 2/7.
Nonfiction : Fiction = 49 : 2.
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Let x stand for the number of minutes spent waiting in line for a rollercoaster at an amusement park. 81 people are sampled at a time. The sample mean is 18 minutes and the sample standard deviation is 0. 5 minutes.
The population mean (μ) is estimated to be: μ = 18 minutes
In this problem, we are given a sample of 81 people who waited in line for a rollercoaster at an amusement park. We are also given the sample mean (18 minutes) and the sample standard deviation (0.5 minutes).
To estimate the population mean (μ), we use the sample mean (18 minutes). Since the sample mean is a good estimate of the population mean, the estimated population mean (μ) is 18 minutes.
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The expected population mean () is: = 18 minutes.
Additionally, the sample mean (18 minutes) and sample standard variation are provided (0.5 minutes).
We use the sample mean to determine the population mean (). (18 minutes). The expected population mean () is 18 minutes because the sample mean is a reliable indicator of the population mean.
The dot plot represents the distribution of wages earned during a one-week period by 12 collage students.
From the line dot plot of 12 college students wages,
The Mean is $118.25
The Median is $118.
What is a mean?It is the average value of the set given.
It is calculated as:
Mean = Sum of all the values of the set given / Number of values in the set
We have,
From the dot plot.
Weekly wages = 112, 113, 116, 116, 116, 118, 118, 119, 119, 124, 124, 124.
Now,
Mean = Sum of all the values / Number of values
= (112 + 113 + 116 + 116 + 116 + 118 + 118 + 119 +119 + 124 +124 +124) / 12
= 118.25
Median = Mid value from the wages.
= 118
Thus,
The Mean is $118.25
The Median is $118.
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select the correct answer .
Which piecewise function is shown on the graph?
A piecewise function which is shown on the graph include the following: A. piecewise function A.
What is a graph?In Mathematics, a graph can be defined as a type of chart that is typically used for the graphical representation of ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-coordinate (x-axis) and y-coordinate (y-axis) respectively.
Based on the graph shown above, we can evaluate the piecewise function based on the following domains (input values or independent values);
For the domain x ≤ -2, the value of y (y-value) is equal to 5. Additionally, for the domain -2 < x < 1,we can logically deduce that the function is a polynomial, which implies that it must have this ordered pair (0, -5):
For -2 < x < 1, when x is equal to 0 and y is equal to -5;
y = x² ± 5
For the domain x ≥ 1, we know that, when y is equal to 0, x would lie between 3 and 4;
y = 2^{x - 2} - 3
When y = 0, the value of x is given by:
0 = 2^{x - 2} - 3
2^{x - 2} = 3
x = 3.584.
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Describe all solution of Ax = b, where A = [2 -5 3 -2 6 -5 -4 7 0] and b = [-3 4 3] Describe the general solution in parametric vector form.
The system has no solution, there is no particular solution x₀, and the general solution does not exist in this case.
The equation Ax = b, where A is a 3x3 matrix and b is a 3x1 column vector, represents a system of linear equations. To find all solutions of this system, we can either use Gaussian elimination or Cramer's rule.
In this case, A = [[2, -5, 3], [-2, 6, -5], [-4, 7, 0]] and b = [[-3], [4], [3]].
Using Gaussian elimination, we can transform the matrix A into its reduced row echelon form and then use back substitution to solve for the variables. The reduced row echelon form of A is:
[[1, 0, 1/2], [0, 1, -5/2], [0, 0, 0]]
Since the last row is all zeros except for the last entry, the system is inconsistent and has no solution.
Alternatively, we can use Cramer's rule to find the solution. The formula for Cramer's rule is:
x_i = det(A_i) / det(A), where A_i is the matrix A with the i th column replaced by the vector b.
In this case, det(A) = 0, so the system has no solution.
The general solution of this system of linear equations is the set of all linear combinations of the columns of A. If the system has a solution, the general solution can be expressed in the parametric vector form as:
x = x₀ + t₁ * v₁ + t₂ * v₂, where x₀ is a particular solution, v₁ and v₂ are the non-trivial solutions, and t₁ and t₂ are scalar parameters.
Since the system has no solution, there is no particular solution x_0, and the general solution does not exist in this case.
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