Sketch the graph of each linear inequality

a) the solution set to A0 can be written as A0 = {(-3z, 0, z) | z ∈ R}

b) the span of the columns of A is a plane in R³ passing through the origin (0, 0, 0).

(a) To find the solution set to the equation A0 = O, where A is a given matrix and O is the zero vector, we need to solve the system of linear equations represented by the augmented matrix [A | O].

Let's perform row operations on the augmented matrix to find the row-echelon form:

Row 2 = (1/2) * Row 2 - Row 1

Row 3 = (1/2) * Row 3 + Row 1

The resulting row-echelon form is:

1 3 9 | 0

0 1 2 | 0

0 0 0 | 0

From this form, we can see that the variables x and z are free variables, while y is a basic variable. Therefore, the solution set can be represented as:

x = -3z

y = 0

z = z

Thus, the solution set to A0 can be written as:

A0 = {(-3z, 0, z) | z ∈ R}

(b) The span of the columns of matrix A does not have the form (i), (ii), or (iii). In this case, the span of the columns of A is a plane in R³ passing through the origin (0, 0, 0). This is because the columns of A are linearly dependent, and therefore, they lie on a common plane in R³.

The fact that the row-echelon form of A has a row of zeros indicates that there is a non-trivial linear combination of the columns that equals the zero vector.  

For more such question on plane . visit :

https://brainly.com/question/30655803

#SPJ8

The probability that three numbers chosen independently and at random from the interval [0,1] are the side lengths of a triangle can be calculated using geometric methods or estimated using simulation or numerical methods.

The probability that three numbers chosen independently and at random from the interval [0,1] are the side lengths of a triangle can be determined by considering the conditions for the triangle inequality theorem to be satisfied.

The triangle inequality theorem states that for a triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the remaining side. In mathematical terms, this can be expressed as:

a + b > c
a + c > b
b + c > a

To calculate the probability, we need to determine the range of values for each side length that satisfies these conditions.

Let's consider the first condition: a + b > c. Since all three side lengths are chosen independently and at random from the interval [0,1], the range of possible values for a, b, and c is from 0 to 1.

If we fix c at 1, then the range of values for a and b that satisfy the condition a + b > c would be 0 < a + b < 2. This forms a triangular region in the 2-dimensional coordinate system.

Similarly, if we fix a at 1, the range of values for b and c that satisfy the condition a + b > c would be 0 < b + c < 2. This forms another triangular region.

Lastly, if we fix b at 1, the range of values for a and c that satisfy the condition a + b > c would be 0 < a + c < 2, forming a third triangular region.

To determine the overall range of valid side lengths that satisfy all three conditions, we need to find the intersection of these three triangular regions.

Considering the areas of the triangular regions, we can calculate the probability by dividing the area of the intersection by the total area of the region defined by the side lengths a, b, and c.

However, calculating the exact probability using geometric methods can be quite complex. Alternatively, we can use simulation or numerical methods to estimate the probability.

For example, we can generate a large number of random sets of three side lengths within the interval [0,1] and check how many of them satisfy the triangle inequality theorem. By dividing the number of valid sets by the total number of generated sets, we can obtain an approximation of the probability.

Keep in mind that the probability of the chosen numbers being the side lengths of a triangle is not 0, but it is also not 1. It falls somewhere in between, and the exact value can be difficult to determine analytically.

Therefore, to summarize, the probability that three numbers chosen independently and at random from the interval [0,1] are the side lengths of a triangle can be calculated using geometric methods or estimated using simulation or numerical methods.

To know more about probability refer here:

https://brainly.com/question/29381779

#SPJ11

The number of students signed up for only the Humanities course is calculated by subtracting n(M∩H), n(L∩H), and n(M∩L∩H) from n(H). Then, we get, the n(H only)=172.

When three sets A, B, and C are given, the union is the set that includes components or things that can either belong to A, B, C, or all three sets. This is given by the formula, \(n(A \cup B \cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B \cap C)\).

Let us denote Math as M, Language Arts as L, and Humanities as H. Then, students taking Math are n(M)=65. Students taking Language arts are n(L)=50. Both Math and Language Arts are n(M∩L)=19, both Math and Humanities are n(M∩H)=5, both Language Arts and Humanities are n(L∩H)=6, and all three courses are n(M∩L∩H)=5.

Also out of 287, 9 did not sign up for any courses. Then, n(M∪L∪H)=287-9=278. Then, the n(H) is calculated as follows,

\(\begin{aligned}n(H)&=n(M\cup L\cup H)-n(M)-n(L)+n(M\cap L)+n(M\cap H)+n(L\cap H)-n(M\cap L \cap H)\\&=278-65-50+19+5+6-5\\&=188\end{aligned}\)

Then, to find n(H only), we get,

\(\begin{aligned}n(H\;\text{only})&=188-n(M\cap H)-n(L\cap H)-n(M\cap L \cap H)\\&=188-5-6-5\\&=172\end{aligned}\)

The required answer is 172.

To know more about the union:

https://brainly.com/question/29055360

#SPJ4

Answer:

C

Step-by-step explanation:

6 is the smallest so the first one would be c because it is the smallest

The correct statement regarding the mode of the sample is given as follows:

a. can be larger, smaller or equal to the mode of the population.

The mode of a data-set is the observation that appears the most often in the data-set.

The mode of the sample is simply the most frequently occurring value in the sample. It is not necessarily expected to be the same as the mode of the population, which is the most frequently occurring value in the entire population.

As a sample is a group taken from an entire population, the observation that appears the most in the sample can be different from the more common observation in the population,

Hence, option (a) is correct.

Learn more about Random sample at:

https://brainly.com/question/29852583

#SPJ4

Therefore , the solution of the given problem of equation comes out to be  9x^3 - 8x^2.

A mathematical equation is a formula that uses the equal sign (=) to connect two statements and express equality. A mathematical statement that proves the equality of two mathematical expressions is known as an equation in algebra. For instance, the components 3x + 5 and 14 in the equation 3x + 5 = 14 are separated by an equal sign. The equal sign is a crucial part of mathematical formulas, especially equations. Algebra is frequently utilized in equations. Algebra is used in mathematics when it is difficult to determine a precise quantity.

Here,

Polynomial function total

Polynomial functions are those that have a leading degree of three or higher. given the total;

(7x^3-4x^2)+(2x^3-4x^2)

Expand => (7x 3-4 times) + (2x 3-4 times)

= > 7x^3-4x^2 + 2x^3-4x^2

assemble similar terms

=> 7x^3 + 2x^3 - 4x^2 -4x^2

=> 9x^3 - 8x^2

Therefore , the solution of the given problem of equation comes out to be 9x^3 - 8x^2.

To know more about equation visit:

https://brainly.com/question/649785

#SPJ1

Answer:

-1

Step-by-step explanation:

to be honest im not sure if its right or not but if u look at the question there are 5 x making it -5 and 6 1's

An exponential model for the population of 20000n is given as P(t) = 20,000(1.144)^t

Suppose

P represents population,

t the number of years of growth and

a, b the constants of the model.

A population numbers 20,000 organisms initially and grows by 14.4% each year.

An exponential model for the population can be written in the form

P = ab^(t).

The initial population is 20,000. We need to find the values of a and b. Since we know the population grows by 14.4% each year, we can write the following equation:

P(t) = 20,000(1 + 0.144)^t

Simplify this equation to get it in the form of P = ab^(t):

P(t) = 20,000(1.144)^t

P(t) = a(1.144)^t

We can see that a = 20,000, so our equation is:

P(t) = 20,000(1.144)^t

This is our exponential model for the population.

To know more about exponential model refer here:

https://brainly.com/question/33502063

#SPJ11

The statements that are true about the function when graphed on a coordinate plane include the following:

C. The relative minimum of the function is (3, 0)

E. When t > 3, the function is increasing.

In order to determine the minimum and maximum of this function, we would have to determine the critical points where the derivative of the function is equal to zero or undefined, and then evaluate the function at these critical points and at the endpoints of the interval.

By taking the first derivative of the given function and factorizing, we have:

f(t) = t³ - 6t² + 9t

f'(t) = 3t² - 12t + 9  

3t² - 12t + 9 = 0

t² - 4t + 3 = 0

(t - 3)(t - 1) = 0

t = 3 and t = 1

Therefore, the critical points of the function are at t = 1 and t = 3.

By taking the second derivative of the given function and factorizing, we have:

f''(t) = 6t - 12

At point t = 1, we have:

f''(1) = 6(1) - 12 = -6 (it is less than zero).

Therefore, f(t) has a local maximum at t = 1.

At point t = 3, we have:

f''(3) = 18 - 12 = 6 (it is greater than zero).

Therefore, the function f(t) has a local minimum at t = 3.

At t = 4, f''(4) = 24 - 12 = 12

At t = 5, f''(5) = 30 - 12 = 18

In conclusion, the relative minimum of the function is (3, 0) and when t > 3, the function would increase.

Read more on function here: https://brainly.com/question/29230332

#SPJ1

Should be 38.89.

Since the number in between 88 and 90 is 99, it will be, or should be, 38.89.

Answer:

128 (character filter filler here)

Step-by-step explanation:

64+8^2 is 128

The given office complex has a total of 1100 employees. Out of these 1100 employees, only some come to work on any given day, while the rest are absent.

If we consider the employees present at work on any given day as the "state", we can create a probability transition matrix that provides information about the probability of the employees being present or absent in the future. We know that if an employee is at work today, there is an 80% chance that she will be at work tomorrow, and if the employee is absent today, there is a 51% chance that she will be absent tomorrow. Hence, we can construct a probability transition matrix for the given scenario as follows:    [1   0]      [0.51 0.8]. Here, the first row represents the probability of remaining absent while the second row represents the probability of remaining present.  We are given that today there are 913 employees at work. We can use the transition matrix to predict the number of employees at work after 5 days.  To do this, we multiply the transition matrix with itself 5 times. We get the following result after multiplying:    

[1   0]      [0.51 0.8] × [1   0]      [0.51 0.8] × [1   0]      [0.51 0.8] × [1   0]      [0.51 0.8] × [1   0]      [0.51 0.8] =     [1.0264 0.0000]      [0.0000 0.9736]

Hence, the number of employees that will be at work after 5 days is:

(1.0264 × 913) = 937.  

The steady-state vector for the given transition matrix can be found by solving the equation P = P × A where P is the steady-state vector and A is the transition matrix. Hence, we have:      

[p1 p2] = [p1 p2] × [1   0]      [0.51 0.8]

We know that p1 + p2 = 1 since the probability of an employee being either present or absent is 1. Substituting p2 = 1 - p1, we get the following equation:  

p1 = 0.51p1 + 0.8(1 - p1)

Solving this equation, we get: p1 = 0.6098 and p2 = 0.3902. Hence, the steady-state vector is [0.6098 0.3902].

The transition matrix for the given scenario is [1 0][0.51 0.8]. The number of employees that will be at work after 5 days is 937. The steady-state vector for the given transition matrix is [0.6098 0.3902].

To learn more about probability transition matrix visit:

brainly.com/question/30654994

#SPJ11

Step-by-step explanation:

2) 212*198 = 41 976

1) 80-37 = 43

9514 1404 393

Answer:

  1. SCR (or PPW). 43 more flavors

  2. GLT. 41,976 notebooks

Step-by-step explanation:

1. Conceptual key: SCR

Lizzy is starting with 37 flavors. She wants a result of 80 flavors, so the change can be computed from ...

  37 + x = 80

  x = 80 -37 = 43

Lizzy needs to create 43 more flavors.

__

Additional comment

This might also be considered to be using the PPW pattern, as both the number she has and the number she creates make up the whole number. The SCR key may better apply to cases where a "rate of change" is involved. There is no rate of change in this problem.

_____

2. Conceptual key: GLT

There are 212 groups of 198 notebooks, so the total is the number of groups times the number in each group.

  (212 boxes)(198 notebooks/box) = 41,976 notebooks

This is true because a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

The area under the probability distribution is a measure of the probability of all the possible outcomes. Since the sum of all the probabilities of all the possible outcomes must be 1, the total area under the probability distribution must also be 1. This is because the probability of an event happening is the area under the curve of the probability distribution at that event. Thus, the total area under the probability distribution must always equal 1.

This is true because a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

Learn more about probability here

https://brainly.com/question/11234923

#SPJ4

Answer:

pi ≈ 3,141593

Step-by-step explanation:

I think this is the right answer

Answer:

2\(\sqrt{23}\)

Step-by-step explanation:

The distance between the points (-4, -7) and (-8, -13) is 2√13 units as per the concept of distance between the two points.

To determine the distance between two points, we can use the distance formula, which is derived from the Pythagorean theorem. The formula is as follows:

\(Distance = \sqrt{((x2 - x1)^2 + (y2 - y1)^2)\)

Given the two points (-4, -7) and (-8, -13), we can assign the coordinates as follows:

x1 = -4, y1 = -7 (coordinates of the first point)

x2 = -8, y2 = -13 (coordinates of the second point)

Now, we substitute these values into the distance formula:

\(Distance = \sqrt{(-8 - (-4))^2 + (-13 - (-7))^2)}\\\\= \sqrt{((-8 + 4)^2 + (-13 + 7)^2}\\= \sqrt{((-4)^2 + (-6)^2)}\)

= √(16 + 36)

= √52

= 2√13

Therefore, the distance between the points (-4, -7) and (-8, -13) is 2√13 units.

To learn more about the distance between two points;

https://brainly.com/question/24485622

#SPJ2

Answer:

x=4.8

Step-by-step explanation:

9.6 is the diameter, and x is the radius. The radius is half of the diameter, so 9.6/2.

x=4.8

Answer:

\(\displaystyle x=\left\{\frac{\pi}{6}+2n\pi, \frac{5\pi}{6}+2n\pi, \frac{3\pi}{2}+2n\pi\right\}, n\in\mathbb{Z}\)

Step-by-step explanation:

We are given the equation:

\(2\cos^2(x)=\sin(x)+1\)

And we want to find all solutions for x.

First, we should put the equation into terms of only one trigonometric ratio.

Since we are given cos²(x), we can turn this into sine. Recall the Pythagorean Identity which states:

\(\sin^2(x)+\cos^2(x)=1\)

Therefore:

\(\cos^2(x)=1-\sin^2(x)\)

By substitution:

\(2(1-\sin^2(x))=\sin(x)+1\)

Distribute:

\(2-2\sin^2(x)=\sin(x)+1\)

Isolate the equation:

\(2\sin^2(x)+\sin(x)-1=0\)

We can factor:

\((2\sin(x)-1)(\sin(x)+1)=0\)

Zero Product Property:

\(2\sin(x)-1=0\text{ or } \sin(x)+1=0\)

Solve for each case:

\(\displaystyle \sin(x)=\frac{1}{2}\text{ or } \sin(x)=-1\)

We can use the unit circle.

sin(x) = 1/2 for every π/6 and 5π/6. So, it will continue every 2π.

sin(x) = -1 every 3π/2. And this will also continue every 2π.

Hence, our solutions are:

\(\displaystyle x=\left\{\frac{\pi}{6}+2n\pi, \frac{5\pi}{6}+2n\pi, \frac{3\pi}{2}+2n\pi\right\}, n\in\mathbb{Z}\)

Note:

If you only need the solutions within the interval [0, 2π), then it is:

\(\displaystyle x=\left\{\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}\right\}\)

Answer:

840

Step-by-step explanation:

7000×2×6 ÷ 100. since it is 2%

= 840

Answer:

       \(\frac{2}{7}\)

Step-by-step explanation:

3.5 = \(\frac{7}{2}\)

swap the numerator and denominator (flip the fraction):

reciprocal of \(\frac{7}{2}\) is \(\frac{2}{7}\)

Answer:

48 people are right handed. 12 are left handed

Step-by-step explanation:

80% of 60 is 48.

The segment AC is the angle bisector of angle BAD and angle BCD.

A ray, segment, or line that divides a given angle into two equal angles is known as an angle bisector.

When something is bisected or bisected, it is split into two equal sections. In geometry, an angle bisector is a line or ray that divides a triangle into two equal angles.

Given that ΔABC ≅ ΔADC Prove: AC bisects ∠BAD and AC bisects ∠BCD​.

From the given data it is concluded that all the corresponding angles of the triangle are equal to each other.

In the triangle ABC and ADC:-

∠BAC = ∠DAC

∠BCA = ∠DCA

From the above proof, it is concluded that the AC bisects ∠BAD and ∠BCD.

Therefore, the segment AC is the angle bisector of angle BAD and angle BCD.

To know more about an angle bisector follow

https://brainly.com/question/24677341

#SPJ1

Answer:

The formula should be the change in y-values with respect to the change in the x-values. She used an incorrect formula

Step-by-step explanation:

The question is incomplete. Here is the complete question.

Anya found the slope of the line that passes through the points (–7, 4) and (2, –3). Her work is shown below. Let (x2, y2) be (–7, 4) and (x1, y1) be (2, –3). m = = = The slope is . What error did she make? She simplified the denominator incorrectly. The denominator simplifies to –7. She labeled the points incorrectly. The point (–7, 4) should be (x1, y1). She used an incorrect formula. The formula should be the change in y-values with respect to the change in the x-values. She used an incorrect formula. The formula should be the sum of the x-values with respect to the sum of the y-values.

Slope of a line is expressed according to the formula;

Slope m = Δy/Δx

Slope = y2-y1/x2-x1

Given the coordinates (–7, 4) and (2, –3), from the coordinates;

x1 = -7, y1 = 4, x2 = 3, y2 = -3

Substitute into the formula;

Slope = -3-4/3-(-7)

Slope = -7/3+7

Slope = -7/10

Based on Anya workings, it can be concluded that Anya used an incorrect formula. The formula should be the change in y-values with respect to the change in the x-values

By what the question asks, I assume that this refers to getting the slope in a linear equation.

The correct option is:

She used an incorrect formula. The formula should be the change in y-values with respect to the change in the x-values.

A general linear equation is given by:

y = a*x + b

where a is the slope and b is the y-intercept.

We know that for a line that passes through points (x₁, y₁) and (x₂, y₂) then the slope can be computed as:

\(a = \frac{y_2 - y_1}{x_2 - x_1}\)

Notice that the formula is the change in y-values with respect to the x-values,

So we can conclude that the correct option is:

"She used an incorrect formula. The formula should be the change in y-values with respect to the change in the x-values."

If you want to learn more, you can read:

https://brainly.com/question/11897796

Answer:

The situation per week is 30 + 6n

where n is the number of weeks

Range of the function is 18 for the first four weeks of data

Step-by-step explanation:

Here, we are to use an arithmetic sequence for a representation.

The first term here is a which is 30 clients while the common difference of d is 6 new clients per week

So we can write the nth term of the sequence as;

Tn = a + (n-1)d

Tn = 30 + (n-1)6

Tn = 30+ 6n -6

Tn = 24 + 6n

where n is the number of weeks

Now for the second week we shall have a total of a+ d clients = 30 + 6 = 36

For the third we have 24 + 6(3) = 24 + 18 = 42

For the fourth, we have 24 + 6(4) = 24 + 24 = 48

Thus the range mathematically means the difference between the highest and lowest for that four weeks = 48-30 = 18

Answer:

A lawn-mowing company is trying to grow its business. It had 18 clients when they started its business and wants to increase by 4 new clients each week.

Use an arithmetic sequence to write a function to represent this real-world situation and determine the range of the function for the first four weeks of data.

f(x) = 4x + 18; 0 ≤ y ≤ 4

29.78% of customers will most likely probability be waiting between 4 and 9 minutes.

The formula for the z score is-

z = (x - μ)/σ

In which,

μ =  mean of 10. 3 minutes

σ = standard deviation of 2. 7 minutes.

z score for the x = 4 minutes.

z = (4 - 10.3)/2.7

z = -2.33

z score for the x = 9 minutes.

z = (9 - 10.3)/2.7

z = -0.48

Thus,

Probability that the customer will have to wait between 4 and 9 minutes.

P(4 < x < 9) = P(-0.48 < z <  -2.33)

Use z score table-

P(4 < x < 9) =  0.3085 - 0.0107

P(4 < x < 9) = 0.2978

P(4 < x < 9) = 29.78%

Thus, 29.78% of customers will most likely be waiting between 4 and 9 minutes.

To know more about the z-score of the normal distribution, here

https://brainly.com/question/21781956

#SPJ4

Answer:

The radical sign

Step-by-step explanation:

In the picture 3 is called the radical sign. So option 3 is correct.