1/2y=5 1/2 help!!!! i don't get it i have to factor it

The z-score corresponding to a sample mean of 84 for a sample size of 9 scores, with a population mean of 80 and a population standard deviation of 12, is 1.

To find the z-score corresponding to a sample mean of m = 84 with a population mean (μ) of 80 and a population standard deviation (σ) of 12, the z-score can be calculated using the formula z = (x - μ) / (σ / √n).

In this case, the population mean (μ) is 80 and the population standard deviation (σ) is 12. The sample mean (m) is given as 84, and the sample size (n) is 9.

To calculate the z-score, we use the formula:

z = (x - μ) / (σ / √n)

Substituting the given values, we have:

z = (84 - 80) / (12 / √9)

Simplifying the expression, we get:

z = 4 / (12 / 3)

z = 4 / 4

z = 1

Therefore, the z-score corresponding to a sample mean of 84 for a sample size of 9 scores, with a population mean of 80 and a population standard deviation of 12, is 1. This indicates that the sample mean is one standard deviation above the population mean. The z-score allows us to compare the sample mean to the population distribution and assess how unusual or typical the sample mean is relative to the population.

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The equation y = 5x/2 represents a proportional relationship with a constant of 5/2.

A proportional relationship is a type of relationship between two quantities in which they maintain a constant ratio to each other.

The equation that defines the proportional relationship is given as follows:

y = kx.

In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.

The equation for this problem is given as follows:

y = 5x/2.

Which is a proportional relationship, as it has an intercept of zero, along with a constant of k = 5/2.

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It is not possible to ascertain the magnitude of angle x without receiving further information.

In order to calculate the magnitude of angle x, we must further data concerning the connection that exists between the points A, B, C, and D. We are unable to draw any judgements regarding the size of angle x because we do not have any additional information.

However, if we have additional information, such as the location of point D in relation to points A, B, and C, then we can use circle theorems to compute the magnitude of angle x. One of the possibilities is that point D lies on the segment of the line AB; in this case, the angle x would be a straight angle, which is equal to 90 degrees. This is because a tangent and a radius always make an angle of 90 degrees with one another at the place where they contact one another. In alternate circumstances in which point D is situated in a different area, the magnitude of angle x will shift proportionately. Therefore, it is not possible to estimate the magnitude of angle x without first determining the precise location of point D or any other information that is pertinent to the problem.

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We first list all the partitions of integers from 1 to 7, then use these lists to compute the values of the partition function p(n) for n from 1 to 7. Therefore, the values of the partition function for integers from 1 to 7 are 1, 2, 3, 5, 7, 11, and 15, respectively.

A partition of a positive integer n is a way of writing n as a sum of positive integers, where the order of the summands does not matter. For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. To compute the partition function p(n), we count the number of partitions of n.

Here are the partitions of integers from 1 to 7:

1: {1}

2: {2}, {1,1}

3: {3}, {2,1}, {1,1,1}

4: {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}

5: {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {1,1,1,1,1}

6: {6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1,1}, {1,1,1,1,1,1}

7: {7}, {6,1}, {5,2}, {5,1,1}, {4,3}, {4,2,1}, {4,1,1,1}, {3,3,1}, {3,2,2}, {3,2,1,1}, {3,1,1,1,1}, {2,2,2,1}, {2,2,1,1,1}, {2,1,1,1,1,1}, {1,1,1,1,1,1,1}

Using this list, we can compute the values of the partition function p(n) for n from 1 to 7:

p(1) = 1

p(2) = 2

p(3) = 3

p(4) = 5

p(5) = 7

p(6) = 11

p(7) = 15

Therefore, the values of the partition function for integers from 1 to 7 are 1, 2, 3, 5, 7, 11, and 15, respectively.

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The correct equation is 0.75w + 0.5w + 2 = 17.

To determine the error in the given equation and find the correct number of laps needed before the sprint to the finish, we can follow these steps:

Step 1: Examine the given equation, w(1.5 + 1)^2 = 17, and identify the error.

The equation provided is incorrect because it does not account for the width and height of the rectangle, as well as the 2-mile sprint to the finish line.

Step 2: Formulate the correct equation based on the given information.

The correct equation should be 0.75w + 0.5w + 2 = 17, where w represents the number of laps. This equation includes the distance covered by riding laps around the block (0.75w + 0.5w) and the additional 2-mile sprint to the finish line.

Step 3: Solve the equation to find the correct number of laps.

By simplifying the equation, we get 1.25w + 2 = 17. Subtracting 2 from both sides gives us 1.25w = 15. Dividing both sides by 1.25 yields w = 12.

Therefore, the correct equation is 0.75w + 0.5w + 2 = 17, and the cyclists will need to ride 12 laps before the sprint to the finish. The given equation and the incorrect number of laps mentioned in the other options do not accurately represent the race distance and the laps required.

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The missing coordinate of point P is sqrt(5/9). The complete coordinates of P in quadrant II are (-2/3, sqrt(5/9)).

To find the missing coordinate of p, we need to use the fact that p lies on the unit circle in the given quadrant. The coordinates of a point on the unit circle are (cosθ, sinθ), where θ is the angle that the point makes with the positive x-axis.
In this case, we know that p lies in quadrant ii, which means that its x-coordinate is negative and its y-coordinate is positive. We also know that the length of the vector OP, where O is the origin and P is the point on the unit circle, is 1.
Using the Pythagorean theorem, we can write:
(OP)^2 = x^2 + y^2 = 1
Substituting the given coordinates of p, we get:
(-2)^2 + 3^2 = 1
4 + 9 = 1
This is clearly not true, so there must be an error in the given coordinates of p.
Therefore, we cannot find the missing coordinate of p using the given information.
Thus, the missing coordinate of point P is sqrt(5/9). The complete coordinates of P in quadrant II are (-2/3, sqrt(5/9)).

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Answer:

No real roots, two complex roots

Step-by-step explanation:

By calculating the discriminant:

[tex]D=b^2-4ac=5^2-4(5)(21)=25-420=-395 < 0[/tex], then there will be no real zeroes. However, there will be two complex roots.

Answer:

missing : refrence

Step-by-step explanation:

In mathematics, an integral is a mathematical object that represents the area between a function and the x-axis on a graph, or the accumulation of a quantity over time.

(a) Integrate by parts, letting dv = 5 − x dx.

Using integration by parts, we can write:

∫x(5-x) dx = x ∫(5-x) dx - ∫[d/dx(x) ∫(5-x) dx] dx

= x [5x - (1/2)x^2] - ∫(0 - (5-x)dx)

= x [5x - (1/2)x^2] - (5x - (1/2)x^2) + C

= - (1/2)x^2 + 10x + C, where C is the constant of integration.

(b) Integrate by substitution, letting u = 5 − x.

Using u-substitution, we can write:

∫x(5-x) dx = ∫(5-u)u du

= ∫(5u - u^2) du

= (5/2)u^2 - (1/3)u^3 + C

= (5/2)(5-x)^2 - (1/3)(5-x)^3 + C, where C is the constant of integration.

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The value of probability is P(A^c∩B^c) = 0.2.

Using the formula P(A) = P(A ∩ B) + P(A ∩ B^c) and P(A^c) = 1 - P(A), we can compute P(A) and P(B) as follows:

P(A) = P(A ∩ B) + P(A ∩ B^c) = 0.3 + 0.15 = 0.45

P(A^c) = 1 - P(A) = 1 - 0.45 = 0.55

Similarly, we can compute P(B) using P(B ∩ A) + P(B ∩ A^c) = P(B ∩ A) + P(A^c ∩ B) = 0.35P, which gives P(B) = 0.35P.

Using the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B), we can compute P(A ∪ B) as follows:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.45 + 0.35P - 0.3 = 0.15 + 0.35P

Since P(A ∪ B) + P(A^c ∪ B^c) = 1, we have

P(A^c ∪ B^c) = 1 - P(A ∪ B) = 1 - (0.15 + 0.35P) = 0.85 - 0.35P

Finally, using the formula P(A^c ∩ B^c) = 1 - P(A ∪ B) = 1 - (0.15 + 0.35P) = 0.85 - 0.35P. Therefore, P(A^c ∩ B^c) = 0.85 - 0.35P.

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The arithmetic sum of the given sequence 4, 9, ..., 219, 224 is 5130.

First, we need to find the common difference (d) between the consecutive terms in this arithmetic sequence. We can do this by subtracting the first term from the second term: 9 - 4 = 5.

Now that we know the common difference, we can determine the number of terms (n) in the sequence using the formula for the last term (L) in an arithmetic sequence: L = a + (n - 1)d, where a is the first term. In this case, the last term (L) is 224, and we have:

224 = 4 + (n - 1)5

Solving for n, we get:

220 = (n - 1)5

n - 1 = 44

n = 45

Now that we have the number of terms, we can compute the sum (S) of the arithmetic sequence using the formula: S = n/2(a + L). Plugging in the values, we get:

S = 45/2(4 + 224)

S = 45/2(228)

S = 45 × 114

S = 5130

So, the arithmetic sum of the given sequence is 5130.

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Displacement is defined as the magnitude and direction of an object's change in position from the starting point.What is displacement?Displacement refers to the overall change in the position of an object over a specified period of time. It takes both magnitude and direction into account.

Displacement, as opposed to distance traveled, is a vector amount that considers not only the total distance traveled but also the direction in which the object moved.

Displacement is the length of the straight line connecting the beginning and ending positions of an object, as well as the direction of this line.

There are a few key things to keep in mind about displacement:Displacement is calculated using the formula: Displacement (Δd) = Final Position - Initial Position (d₂ - d₁)

Displacement is a vector amount since it includes both magnitude and direction.

If an object moves around in a circle and finishes where it began, its displacement will be zero but the distance it travels will not.

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Since our test statistic of 4.5 is greater than the critical value of 2.700, we reject the null hypothesis. Therefore, we say there is enough evidence to reject the manufacturer's claim.

A good way to test if a sample with Variance of 4.3 is worth rejecting by manufacturer, we can use a Chi-Square test with (n-1) degrees of freedom. Where n is the sample size.

null hypothesis: the variance of the population is equal to 8.6

alternative hypothesis: the variance of the population is less than 8.6.

The test statistic is given by:

Chi-Square = (n - 1) * sample variance / population variance

From the problem statement, we have

n = 10

sample variance = 4.3

population variance = 8.6

Substituting these values, we get:

chi-square = (10 - 1) * 4.3 / 8.6 = 4.5

The critical value for a chi-square distribution with 9 degrees of freedom at a significance level of 0.01 is 2.700.

Since our test statistic of 4.5 is greater than the critical value of 2.700, we reject the null hypothesis.

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Answer: 4056

Step-by-step explanation:

take cube root of 17576= 26

26*26*6=4056

i) The percentage of enquires where no sale is made is 42%.

ii) If in a month 2,800 inquiries are made, the company would expect to make sales of 1,624.

The percentage refers to the quotient of a number or value multiplied by 100.

The quotient is the result of a division operation that compares a portion of a quantity with the whole.

The percentage of people who make telephone enquires and buy the company's products = 58%

i) The percentage of the people who make telephone inquiries but do not buy the company's products = 42% (100% - 58%)

ii) The number of inquiries made in a month = 2,800

The expected number of sales for the month = 1,624 (2,800 x 58%)

Thus, based on the percentage of expected sales, when 2,800 inquiries are made, the company should make 1,624 sales.

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A. P(6, then 1) = 1/90

B. P(even, then 5) = 1/18

C. P(8, then odd) = 1/18

D. P(3, then prime) = 2/45

E. P(prime, composite) = 4/15

F. P(even, then 3, then 5) = 1/144

Given:

Total number of cards: 10

A. P(6, then 1):

P(6, then 1) = 1/10 x 1/9

= 1/90

B. P(even, then 5):

Number of favorable outcomes: 5 x 1 = 5

P(even, then 5) = 5/10 x 1/9

= 1/18

C. P(8, then odd):

Number of favorable outcomes: 1 x 5 = 5

P(8, then odd) = 1/10 x 5/9

= 1/18

D. P(3, then prime):

Number of favorable outcomes: 1 x 4 = 4

P(3, then prime) = 1/10 x 4/9

= 2/45

E. P(prime, composite):

Number of favorable outcomes: 4 x 6 = 24

P(prime, composite) = 4/10 x 6/9

= 4/15

F. P(even, then 3, then 5):

Number of favorable outcomes: 5 x 1 x 1 = 5

P(even, then 3, then 5) = 5/10 x 1/9 x 1/8

= 1/144

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The simplified ratio is:    

1 : 2 : 6 or 1 : 2/1 : 6/1.

The question requires us to determine the ratio of the earnings of the three children in the Skosana family. Given that the eldest, middle and youngest children earn R270, R180 and R90 respectively, we can express these amounts as a ratio as follows:

R270 : R180 : R90

Let's simplify this ratio by dividing each amount by the highest common factor of 90.

R270 ÷ 90 = 3

R180 ÷ 90 = 2

R90 ÷ 90 = 1

Therefore, the ratio of the earnings of the three children in the Skosana family is:3 : 2 : 1.

An explanation of how we came up with the answer is as shown below:

We can express the eldest child's earning as a ratio of the total earnings as follows:

R270 : (R270 + R180 + R90)

Simplifying this ratio, we get:

3R : (3R + 2R + R) = 3R : 6R

= 1 : 2

Similarly, we can express the middle child's earning as a ratio of the total earnings as follows:

R180 : (R270 + R180 + R90)

Simplifying this ratio, we get:

2R : (3R + 2R + R) = 2R : 6R

= 1 : 3

Finally, we can express the youngest child's earning as a ratio of the total earnings as follows:

R90 : (R270 + R180 + R90)

Simplifying this ratio, we get:

R : (3R + 2R + R) = R : 6R

= 1 : 6

Therefore, the ratio of the earnings of the three children in the Skosana family is:

1 : 2 : 6.

However, we can further simplify this ratio by dividing each amount by the highest common factor of

6.1 ÷ 1 = 12 ÷ 2

= 36 ÷ 6

= 1

Therefore, the simplified ratio is:

1 : 2 : 6 or 1 : 2/1 : 6/1.

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There is an 88% chance that he will complete his next free throw.

Now, When Steph Curry has an 88% free throw rating, that means on average he makes 88 free throws out of 100 attempts.

Now, let's look at the probability of him making 16 free throws in a row.

Since each free throw is independent of the others, the probability of making 16 in a row is simply 0.88 to the power of 16

since he has an 88% chance of making each one).

That comes out to about, 0.284, or 28.4% chance of making 16 in a row.

Therefore, the probability that he completes his next free throw after making 15 in a row is still 88%,

since each free throw is independent of the others.

So, there is an 88% chance that he will complete his next free throw.

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df(0) = [0 0 0; 0 0 0]

d(g o f)(0) = [0 0; 0 0].

We have f: R^3 → R^2 and g: R^2 → R^2.

Using the chain rule, we have:

d(g o f)(0) = dg(f(0)) ◦ df(0)

First, let's find df(0):

df(0) = [∂f₁/∂x₁(0) ∂f₁/∂x₂(0) ∂f₁/∂x₃(0); ∂f₂/∂x₁(0) ∂f₂/∂x₂(0) ∂f₂/∂x₃(0)]

We know that f(0) = (0, 0, 0) and f(0) = (1, 2), so:

f₁(0) = 1, f₂(0) = 2

∂f₁/∂x₁(0) = ∂f₁/∂x₂(0) = ∂f₁/∂x₃(0) = 0

∂f₂/∂x₁(0) = ∂f₂/∂x₂(0) = ∂f₂/∂x₃(0) = 0

Next, let's find dg(f(0)):

dg(x, y) = [∂g₁/∂x ∂g₁/∂y; ∂g₂/∂x ∂g₂/∂y]

dg(1, 2) = [2 1; 6 3]

Finally, we can find d(g o f)(0):

d(g o f)(0) = dg(f(0)) ◦ df(0) = [2 1; 6 3] ◦ [0 0 0; 0 0 0] = [0 0; 0 0]

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The derivative of the composition g o f at (0) given by applying the chain rule is [2 4; 3 1].

The problem requires finding the derivative of the composition g o f at (0).

Using the chain rule, we can express this derivative as the product of the Jacobian matrix of g with respect to its inputs and the Jacobian matrix of f with respect to its inputs, evaluated at (0).

The Jacobian matrix of g is given by:

[ 2y 1 2x ]

[ 3y 3x ]

If T : Rn → R

m is a linear transformation, then T(0) = 0.

Evaluating this at f(0) = (1, 2) gives:

[ 4 2 ]

[ 6 3 ]

The Jacobian matrix of f is given by:

[ 1 0 0 ]

[ 0 1 0 ]

Evaluating this at 0 gives:

[ 1 0 0 ]

[ 0 1 0 ]

Multiplying these two matrices, we get:

[ 2 4 ]

[ 3 1 ]

Therefore, d(g o /)(0) = [2 4; 3 1].

In summary, we used the chain rule to find the derivative of the composition g o f at (0), which is given by the product of the Jacobian matrix of g and the Jacobian matrix of f, both evaluated at the same point.

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As the population (t) grows, the probability of no birthday collision reduces. This is due to the fact that as the population grows, the likelihood of two or more people having the same birthday rises.

The probability of no birthday collision among t people can be computed using the formula:

P(no collision) = 1 x (364/365) x (363/365) x ... x [(365-t+1)/365]

For t = 10, we have:

P(no collision) = 1 x (364/365) x (363/365) x ... x (356/365)
P(no collision) = 0.883
Therefore, the probability of no birthday collision among 10 people is 0.883 or approximately 88.3%.

For t = 25, we have:

P(no collision) = 1 x (364/365) x (363/365) x ... x (341/365)
P(no collision) = 0.568
Therefore, the probability of no birthday collision among 25 people is 0.568 or approximately 56.8%.

For t = 40, we have:

P(no collision) = 1 x (364/365) x (363/365) x ... x (326/365)
P(no collision) = 0.108
Therefore, the probability of no birthday collision among 40 people is 0.108 or approximately 10.8%.

In general, the probability of no birthday collision decreases as the number of people (t) increases. This is because the likelihood of two or more people sharing the same birthday increases as the number of people increases.

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Lisa needs 2 pints of blue paint and 4 pints of white paint.

To paint her bedroom walls, Lisa needs a total of 6 pints of blue paint and white paint.

One-fourth (1/4) of this quantity is blue paint and the rest is white paint. We have to find what amount of blue paint and white paint Lisa need.

The total quantity of paint Lisa needs to paint her bedroom is 6 pints.

Let B be the quantity of blue paint Lisa needs.

Then the quantity of white paint she needs is 6 - B (since one-fourth of the total quantity is blue paint).

Hence, B + (6 - B) = 64B + 6 - B = 24B = 2

Therefore, Lisa needs 2 pints of blue paint and (6 - 2) = 4 pints of white paint. (Here, the total quantity of paint is taken as 24 units in order to avoid fractions).

Lisa needs 2 pints of blue paint and 4 pints of white paint.

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By using the multiplication concept, we found that 1/5 of 30 is equal to 6. The following problem can be solved by multiplying 1/5 × 30. It is one of the fundamental arithmetic operations.

The multiplication 1/5 × 30 is used to solve the problem of finding the result when 1/5 of 30 is taken. Multiplication is a fundamental arithmetic operation taught to students in the early grades. Multiplication can be used to solve a variety of mathematical problems, including those that involve finding the total value of multiple items or the number of items in a set. In this case, the multiplication 1/5 × 30 is used to solve the problem of finding the result when 1/5 of 30 is taken.

To find the result of 1/5 of 30, we must multiply 30 by 1/5. To multiply a fraction by a whole number, we can multiply the numerator of the fraction by the whole number and then divide the result by the denominator of the fraction. So,

= 1/5 × 30

= (1 × 30)/5

= 30/5

= 6

Therefore, the result of 1/5 of 30 is 6. This means that if we divide 30 into five equal parts, each part will have a value of 6. The multiplication 1/5 × 30 can solve the problem of finding the result when 1/5 of 30 is taken. By using the multiplication formula, we found that 1/5 of 30 is equal to 6.

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Answer:

Step-by-step explanation:

A squiggly line can refer to a variety of different shapes or patterns. To create an equation using cosine that resembles a squiggly line, you can use a combination of sine and cosine functions with different frequencies and amplitudes. Here's an example equation that produces a squiggly pattern:

y = A * cos(B * x) + C * sin(D * x)

In this equation, A, B, C, and D are constants that you can adjust to modify the shape and characteristics of the squiggly line. By experimenting with different values for these constants, you can create various squiggly patterns. Keep in mind that the specific equation may vary depending on the exact shape and features you have in mind for the squiggly line.

The random variable x follows an exponential distribution with a parameter λ = 3. This distribution is commonly used to model the time between events occurring at a constant average rate.

The exponential distribution is characterized by its probability density function (PDF) and cumulative distribution function (CDF).

In the exponential distribution, the parameter λ represents the rate parameter or the average number of events occurring per unit of time. In this case, with λ = 3, we can interpret it as an average of 3 events occurring per unit of time.

The PDF of the exponential distribution with parameter λ is given by f(x) = λe^(-λx), where x is a non-negative value. This function describes the probability of observing a specific value of x.

The CDF of the exponential distribution is given by F(x) = 1 - e^(-λx). It represents the probability that x is less than or equal to a given value.

The exponential distribution is widely used in various fields such as reliability analysis, queueing theory, and survival analysis. It is particularly useful when modeling the time between events with a constant average rate.

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Option B suggests that the answer options could be interpreted differently by different users. This could lead to inconsistencies in how respondents rate the variety of material available. Different interpretations of the rating scale or varying perceptions of what constitutes a high or low variety could impact the survey results.

Option C states that the survey is biased because it is being taken only by the service's users. This introduces a potential sampling bias since the survey is limited to the service's user base. The opinions and experiences of non-users are not included, which may not provide a comprehensive understanding of the variety of material available. The results may be skewed towards the preferences and perspectives of the service's existing users.

Option A and Option D are not directly related to potential influences on the survey results. Option A addresses who can take the survey, but it does not pertain to the potential biases or variations in responses. Option D discusses the mode of survey administration, but it does not specifically address factors that could affect the survey results themselves.

Therefore, options B and C are the choices that could affect the results of the survey.

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Based on the given scatterplot, the trend appears to be a negative association between age and GPA. As age increases, GPA tends to decrease.

In a scatterplot, the trend represents the general pattern or direction of the relationship between two variables. In this case, the variables are age and GPA. The scatterplot shows that as age increases, there is a general tendency for GPA to decrease. This suggests a negative association between the two variables.

There could be several reasons for this negative association. It could be that older students have more responsibilities and less time to devote to their studies, leading to lower GPAs. Alternatively, it could be that older students are more likely to have completed more difficult courses earlier in their college careers, leading to lower GPAs in subsequent courses.

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The length of the line segment that connects the points (4, -1) and (9, 7) is approximately 9.43 units.

The distance formula used in finding the distance between two points is expressed as;

[tex]d = \sqrt{(x_2 - x_1)^2+( y_2 - y_1)^2}[/tex]

Given that; the coordinates are (4, -1) and (9, 7), so we have:

x₁ = 4

y₁ = -1

x₂ = 9

y₂ = 7

Substituting these values into the distance formula, we get:

[tex]d = \sqrt{(x_2 - x_1)^2+( y_2 - y_1)^2}\\\\d = \sqrt{(9 - 4)^2+( 7 - (-1))^2}\\\\d = \sqrt{(5)^2+( 7 + 1)^2}\\\\d = \sqrt{(5)^2+( 8)^2}\\\\d = \sqrt{ 25 + 64}\\\\d = \sqrt{ 89}\\\\d = 9.43[/tex]

Therefore, the length of the line segmnet is 9.43 units.

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The general solution is x(t) = c1[cos(t), sin(t)] + c2[cos(t), -sin(t)].

The general solution of a differential equation x'(t) = Ax(t), with matrix A = [0 -1; 1 0], can be found by determining the eigenvalues and eigenvectors of the matrix A.

For this matrix, the eigenvalues are λ1 = i and λ2 = -i. The corresponding eigenvectors are x₁= [1, i] and x₂ = [1, -i].

The general solution of the differential equation is given by the linear combination of the eigenvector solutions:

x(t) = c₁x₁(t) + c₂x₂(t), where c₁ and c₂ are constants.

The solutions x₁(t) and x₂(t) can be expressed as:

x₁(t) = [cos(t), sin(t)] x₂(t) = [cos(t), -sin(t)]

Thus, the general solution is x(t) = c₁[cos(t), sin(t)] + c₂[cos(t), -sin(t)].

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The general solution of the given system is x(t) = c1t(1 1) + c2t(1 -1).

To find the general solution of x′ = (−1 1 −1 −1 ) x, we first need to find the eigenvalues and eigenvectors of the matrix A = (−1 1 −1 −1).

The characteristic polynomial of A is given by det(A - λI) = 0, where I is the 2x2 identity matrix and λ is the eigenvalue:

|−1-λ 1 |

|-1 -λ| = (-1-λ)(-1-λ) - (-1)(1) = λ^2 + 2λ = λ(λ+2) = 0.

So the eigenvalues of A are λ1 = 0 and λ2 = -2.

To find the eigenvectors corresponding to each eigenvalue, we need to solve the equations:

(A - λ1I)x1 = 0 and (A - λ2I)x2 = 0.

For λ1 = 0, we have:

(A - λ1I)x1 =

| -1 1 |

| -1 -1 | x1 = 0.

Solving this system of equations, we get x1 = t(1 1), where t is any scalar.

For λ2 = -2, we have:

(A - λ2I)x2 =

| 1 1 |

| -1 -3 | x2 = 0.

Solving this system of equations, we get x2 = t(1 -1), where t is any scalar.

Thus, the general solution of x′ = (−1 1 −1 −1 ) x can be written as a linear combination of the eigenvectors:

x(t) = c1t(1 1) + c2t(1 -1),

where c1 and c2 are constants that depend on the initial conditions.

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If W = U ⊕ V, then U ∩ V = {0} which can be proved by proving {0} is an element of U ∩ V and there are no other elements in U ∩ V besides {0} for the vector space.

To show that if W = U ⊕ V, then U ∩ V = {0}, we need to prove two things:

1. {0} is an element of U ∩ V.
2. There are no other elements in U ∩ V besides {0}.

Step 1: Show that {0} is an element of U ∩ V.

Since U and V are subspaces of the vector space W, they both must contain the zero vector (0) as per the definition of a subspace. Therefore, the zero vector is in both U and V, which implies that 0 is an element of U ∩ V.

Step 2: Show that there are no other elements in U ∩ V besides {0}.

Suppose there is a nonzero vector x that belongs to U ∩ V. This means x is in both U and V. Since W = U ⊕ V, any vector in W can be uniquely written as the sum of a vector from U and a vector from V. Thus, x can be written as:

x = u + v

where u is a vector from U and v is a vector from V. However, x is also in both U and V, so we can rewrite the equation as:

x = x + 0

Since the sum of vectors from U and V is unique, we must have u = x and v = 0. But this contradicts our initial assumption that x is a nonzero vector, as x ∈ V and we assumed x ≠ 0. Therefore, there can be no other elements in U ∩ V besides {0}.

In conclusion, if W = U ⊕ V, then U ∩ V = {0}.

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