Help Please, will give brainly if correct

The complete solution of the given initial value problem  can be given by,

y(t) = [tex]\frac{7}{4} e^{3t} +\frac{7}{12} e^{-t} + -2te^{2t} -\frac{4}{3} e^{2t}[/tex]

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable. For example, dy/dx = 5x.

Therefore, the complete solution of the given  initial value problem  can be given by,

y(t) = [tex]\frac{7}{4} e^{3t} +\frac{7}{12} e^{-t} + -2te^{2t} -\frac{4}{3} e^{2t}[/tex]

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The full solution to the initial value problem can be provided by,

y(t) = 7/4e^3t + 7/12 e^ -t - 2e^2t- 4/3e^2t

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable. For example, dy/dx = 5x.

As a result, the full solution to the initial value problem can be provided by, y(t) =  7/4e^3t + 7/12 e^ -t - 2e^2t- 4/3e^2t

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The value of people in this proportion is equal to 165/10 or 10.67.

In Mathematics, a proportion can be defined as an equation which is typically used to represent (indicate) the equality of two ratios. This ultimately implies that, proportions can be used to establish that two ratios are equivalent and solve for all unknown quantities.

By applying direct proportion to the given information, we have the following mathematical expression:

3/10/p = 4/5/1/4

3/10 × p = 4/5 × 4

3p/10 = 16/5

Cross-multiplying, we have the following:

15p = 160

People, p = 160/15

People, p = 10.67

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The monument is made up of 150,124 blocks.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

Monument:

50,123 sandstone blocks

100,001 limestone blocks.

Now,

The total number of blocks.

= 50,123 + 100,001

= 150124

Thus,

150,124 blocks.

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The solution to the system of equations is x = 0, y = 3, and z = 4.

What is algebraic equation ?

An algebraic equation or polynomial equation is an equation of the form P=0 where P is a polynomial with coefficients in some field, often the field of the rational numbers.

First, multiply the first equation by 2 and add it to the second equation to eliminate x:

-3x - 5y + z = -8

2 * (-3x - 5y + z) = 2 * -8

-6x - 10y + 2z = -16

2x + 8y - z = -4

-6x - 10y + 2z + 2x + 8y - z = -16 - 4

-4x - 2y + 3z = -20

Next,  divide the third equation by 3 to obtain y:

x + 2y - z = 2

y = (2 - x + z) / 2

x + 2y - z = 2

x + 2 * (2 - x + z) / 2 - z = 2

x + 2 - x + z - z = 2

z = 4 - x,

y = (2 - x + 4 - x) / 2

y = (2 - 2x + 4) / 2

y = (6 - 2x) / 2

z = 4 - x

4 = 4 - x

x = 0

Plugging in the values of x and z into the equation for y:

y = (6 - 2x) / 2

y = (6 - 2 * 0) / 2

y = 6 / 2

y = 3

So, the solution to the system of equations is x = 0, y = 3, and z = 4.

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The solution to the system of equations for all three variables is x = -3, y = -5, and z = -15.

We can solve this system of equations using the elimination method or substitution method. Here, I will use the elimination method:

Starting with the first equation:

-3x - 5y + z = -8

Adding the second equation:

-3x - 5y + z + 2x + 8y - z = -8 + (-4)

-x + 3y = -12

Adding the third equation:

-x + 3y + x + 2y - z = -12 + 2

2y = -10

Solving for y:

y = -5

Substituting y back into the first equation:

-3x - 5(-5) + z = -8

-3x + 25 + z = -8

-3x + z = -33

Substituting y back into the second equation:

2x + 8(-5) - z = -4

2x - 40 - z = -4

2x - z = 36

Adding the two equations to eliminate x:

-3x + z + 2x - z = -33 + 36

-x = 3

Solving for x:

x = -3

Substituting x and y back into the third equation:

-3 + 2(-5) - z = 2

-3 - 10 - z = 2

-13 - z = 2

Solving for z:

z = -15

Therefore, the solution is x = -3, y = -5, and z = -15.

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Answer:

Hence the correct answer is 256

Two ratios have proportional relationship.

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.

A ratio is a mathematical expression written in the form of a:b, where a and b are any integers.

Given question,

What is the relationship between the ratios?

Ratio defines the relationship between the quantities of two or more objects. It is used to compare the quantities of the same kind. If two or more ratios are equal, then it is said to be in proportion.

Hence, two ratio have the proportional relationship.

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The empirical formula of a compound is the simplest and lowest whole number ratio of the atoms of each element that makes up a compound.

To find the empirical formula from percentages, you need to first calculate the mass of each element present in the compound, based on the percentage of each element. Once you have the mass of each element, divide each mass by the atomic mass of that element to get the number of moles of each element. Then divide each mole by the smallest number of moles to get the mole ratio. This ratio is the empirical formula for the compound.

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The sum can be used when sum gives the situations in which the probability of a union of events can be calculated by summing probabilities together. and product rule  If there are n(A) ways to do A and n(B) ways to do B, then the number of ways to do A and B is n(A) × n(B)

What is Probability?

Probability is an area of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.

Reason:

The probability rule of sum describes scenarios in which the probability of a union of events can be computed by adding probabilities. It is frequently used on mutually exclusive occurrences, which are events that cannot occur at the same moment.

If there are n(A) ways to perform A and n(B) ways to do B, then the total number of ways to accomplish A and B is n(A) n. (B). This is true if the number of possible ways to do A and B are independent; the number of options for doing B is the same regardless of the option you chose for A.

The probability product rule states that P(AnB) = P(A) * P (B)

This implies that the events must be distinct.

In contrast, for the probability sum rule.

P(A + B) = P(A) + P(B) (B)

This suggests that P(AnB) = 0, indicating that they are disjoint or mutually exclusive.

Box 2 states that the sum rule is used when events are independent, however this is not correct; it is employed when events are mutually exclusive.

Two events are said to be mutually exclusive in probability theory if they cannot occur at the same time or concurrently. In other words, discontinuous events are those that are mutually exclusive. If two occurrences are regarded discontinuous, the likelihood of both occurring at the same time is zero.

If A and B are the two events, then the probability of their disjoint is stated as:

P (A and B) = 0 for the probability of a disjoint (or mutually exclusive) event.

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The sum can be used when sum gives the situations in which the probability of a union of events can be calculated by summing probabilities together. and product rule  If there are n(A) ways to do A and n(B) ways to do B, then the number of ways to do A and B is n(A) × n(B)

What is Probability?

Probability is an area of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.

Reason:

The probability rule of sum describes scenarios in which the probability of a union of events can be computed by adding probabilities. It is frequently used on mutually exclusive occurrences, which are events that cannot occur at the same moment.

If there are n(A) ways to perform A and n(B) ways to do B, then the total number of ways to accomplish A and B is n(A) n. (B). This is true if the number of possible ways to do A and B are independent; the number of options for doing B is the same regardless of the option you chose for A.

The probability product rule states that P(AnB) = P(A) * P (B)

This implies that the events must be distinct.

In contrast, for the probability sum rule.

P(A + B) = P(A) + P(B) (B)

This suggests that P(AnB) = 0, indicating that they are disjoint or mutually exclusive.

Box 2 states that the sum rule is used when events are independent, however this is not correct; it is employed when events are mutually exclusive.

Two events are said to be mutually exclusive in probability theory if they cannot occur at the same time or concurrently. In other words, discontinuous events are those that are mutually exclusive. If two occurrences are regarded as discontinuous, the likelihood of both occurring at the same time is zero.

If A and B are the two events, then the probability of their disjoint is stated as:

P (A and B) = 0 for the probability of a disjoint (or mutually exclusive) event.

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So, the values of the trigonometric functions of t are sin(t) = 1/7, cos(t) = -√(1 - (1/7)^2) and tan(t) = -(1/7)/(-√(1 - (1/7)^2)).

Given that csc(t) = 7 and cos(t) < 0, we can find the values of the other trigonometric functions of t using the relationships between the trigonometric functions.

csc(t) = 1/sin(t), so sin(t) = 1/7

Since cos(t) < 0, t is in the second or third quadrant, where sin(t) is positive. Hence, t is in the third quadrant.

Therefore, the values of the trigonometric functions of t are:

sin(t) = 1/7

cos(t) = -√(1 - sin^2(t)) = -√(1 - (1/7)^2)

tan(t) = sin(t)/cos(t) = -(1/7)/(-√(1 - (1/7)^2))

Note that the values of the other trigonometric functions, such as cot(t), sec(t), and csc(t), can be found using the definitions of these functions and the values of sin(t) and cos(t) that we have found.

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So, the values of the trigonometric functions of t are  [tex]Sin(t)=\frac{1}{7}[/tex], [tex]Cos(t)=-\sqrt{[1-(\frac{1}{7})^2] }[/tex]  and  [tex]tan(t)=\frac{\frac{1}{7} }{-\sqrt{1-(\frac{1}{7})^2 } }[/tex].

Given that csc(t) = 7 and cos(t) < 0, we can find the values of the other trigonometric functions of t using the relationships between the trigonometric functions.

[tex]Cosec(t)=7[/tex]

We known

⇒ [tex]Sin(t)=\frac{1}{Cosec(t)} =\frac{1}{7}[/tex]

Since cos(t) < 0, t is in the second or third quadrant, where sin(t) is positive. Hence, t is in the second quadrant.

Therefore, the values of the trigonometric functions of t are:

⇒ [tex]Sin(t)=\frac{1}{7}[/tex]

⇒ [tex]Cos(t)=-\sqrt{[1-(Sin^2t)]}[/tex]

⇒ [tex]Cos(t)=-\sqrt{[1-(\frac{1}{7})^2] }[/tex]

⇒ [tex]tan(t)=\frac{Sin(t)}{Cos(t)}[/tex]

⇒ [tex]tan(t)=\frac{\frac{1}{7} }{-\sqrt{1-(\frac{1}{7})^2 } }[/tex]

∴ Note that The definitions of the other trigonometric functions, along with the values of Sin(t) and Cos(t), can be used to determine the values of cot(t), Sec(t), and Cosec(t).

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The number of people on the escalator during rush hour will depend on how long rush hour lasts, so it is not possible to provide a single answer without knowing the length of rush hour.

At rush hour, two people per second step onto the escalator, so in one second, 2 people get on the escalator.

The escalator takes 20 seconds to bring a person from the bottom to the top, so for every 20 seconds, 2 people get on the escalator and 2 people get off the escalator.

To find the number of people on the escalator during rush hour, we need to determine the number of people on the escalator after a certain amount of time, say t.

The number of people on the escalator at time t can be calculated as follows:

people on the escalator at time t = 2 * t - 2 * floor(t / 20)

where floor(t / 20) represents the number of complete cycles (20 seconds each) that have occurred.

For example, if t = 40 seconds, then floor(t / 20) = 2, so the number of people on the escalator would be:

people on the escalator at time 40 = 2 * 40 - 2 * 2 = 76

The number of people on the escalator will continue to increase as long as rush hour continues, so it is not possible to give a single answer to the question "how many people are on the escalator during rush hour?" without more information.

In general, the number of people on the escalator during rush hour will depend on how long rush hour lasts, so it is not possible to provide a single answer without knowing the length of rush hour.

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The number of people on the escalator during rush hour will depend on how long rush hour lasts, so it is not possible to provide a single answer without knowing the length of rush hour.

At rush hour, two people per second step onto the escalator, so in one second, 2 people get on the escalator.

The escalator takes 20 seconds to bring a person from the bottom to the top, so for every 20 seconds, 2 people get on the escalator and 2 people get off the escalator.

To find the number of people on the escalator during rush hour, we need to determine the number of people on the escalator after a certain amount of time, say t.

The number of people on the escalator at time t can be calculated as follows:

people on the escalator at time t = 2 * t - 2 * floor(t / 20)

where floor(t / 20) represents the number of complete cycles (20 seconds each) that have occurred.

For example, if t = 40 seconds, then floor(t / 20) = 2, so the number of people on the escalator would be:

people on the escalator at time 40 = 2 * 40 - 2 * 2 = 76

The number of people on the escalator will continue to increase as long as rush hour continues, so it is not possible to give a single answer to the question "how many people are on the escalator during rush hour?" without more information.

In general, the number of people on the escalator during rush hour will depend on how long rush hour lasts, so it is not possible to provide a single answer without knowing the length of rush hour.

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After determining the mean the variance is 1.5689 and standard deviation is 1.253.

The values are 7.2, 7.6, 8.5, 8.7, 9.0, 9.3, 9.4, 10.2, 10.9, 11.3.

Firstly we find the mean of the numbers.

Mean = Sum of the Numbers/Total numbers

Mean = (7.2 + 7.6 + 8.5 + 8.7 + 9.0 + 9.3 + 9.4 + 10.2 + 10.9 + 11.3)/10

Mean = 92.1/10

Mean = 9.21

Now we determine the variance.

The formula of variance is:

x                             x - Mean                                     (x - Mean)^2

7.2                         (7.2-9.21) = -2.01                                4.0401

7.6                         (7.6-9.21) = -1.61                                 2.5921

8.5                         (8.5-9.21) = -0.71                                0.5041

8.7                          (8.7-9.21) = -0.51                                0.2601

9.0                          (9.0-9.21) = -0.21                               0.0441

9.3                          (9.3-9.21) = 0.09                                0.0081

9.4                          (9.4-9.21) = 0.19                                 0.0361

10.2                         (10.2-9.21) = 0.99                              0.9801

10.9                         (10.9-9.21) = 1.69                                2.8561

11.3                          (11.3-9.21) = 2.09                                 4.3681

Total =                                                                                 15.689

Variance = ∑(x - Mean)^2/10

Variance = 15.689/10

Variance = 1.5689

Standard Deviation = √Variance

Standard Deviation = √1.5689

Standard Deviation = 1.253

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To make a kite shape, we need two pairs of equal length sticks and a diagonal connecting the midpoints of each pair.

We can choose the following pairs of equal length sticks from the given pieces of cardboard: (3 ft, 3 ft), (6 ft, 6 ft), or (9 ft, 9 ft).

For the diagonal stick, we can choose the longest piece of cardboard, which is 9 ft.

Therefore, we can make a kite shape with dimensions (3 ft, 3 ft, 9 ft) or (6 ft, 6 ft, 9 ft) or (9 ft, 9 ft, 9 ft).

The resulting shape will have two sticks of equal length forming the top and bottom of the kite, and the longest stick forming the diagonal.

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From the cardboard pieces provided, we can select the following sets of equal-length sticks: (3 ft, 3 ft), (6 ft, 6 ft), or (9 ft, 9 ft).

We can use the longest 9-foot length of cardboard as the diagonal stick.

So, using the measurements (3 feet, 3 feet, 9 feet), (6 feet, 6 feet, 9 feet), or (9 ft, 9 ft, 9 ft).

Two sticks of equal length will form the top and bottom of the resulting shape, with the longest stick making the diagonal.

Answer: 72.5

Step-by-step explanation:

120% = 120/100 = 1.2

1.2*x = 87

x = 87/1.2 = 72.5

The following limits a) the limit does not exist. b) the limit does not exist and c) the limit does not exist.

a) limx→5+ 1/x−5 = 1/0 which is undefined. Therefore, the limit does not exist.

b) limx→5− 1/x−5 = −1/0 which is undefined. Therefore, the limit does not exist.

c) The two-sided limit of 1/x−5 as x approaches 5 from both the left and the right side is undefined. Therefore, the limit does not exist.

A limit in mathematics is a point at which a function approaches the output for the specified input values. Calculus and mathematical analysis both rely heavily on limits, which are also needed to describe concepts like continuity, derivatives, and integrals.

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There are 149.91 pounds in 68 kilogram, or 68 kilogram weighs 149.91 pounds.

The Price Per Kilogram (Pk) shall replace the Price Per Pound (Pp) (Kg)

Just divide the kg price by 2.2046 to get the lb price to convert cost per kilogramme to cost per pound.

In principle, two pounds are equal to one kilogramme (there is a longer decimal position, but I abbreviate it to 2.2046).

The weight of a kilogramme is roughly 2.2046226218 pounds. The technique can be used to convert kilogrammes to pounds and is relatively simple. The value of a kilogramme merely needs to be multiplied by 2.2046226218.

68 kg weights 149.91 pounds, or 68 kg is 149.91 pounds. Conversion of wages 149.91434 pounds (lb), 68 kilogrammes (kg)

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To find the order and degree of a given differential equation, simply count the highest derivative and the highest power of the unknown function that appear in the equation.

What is a differential equation?

A differential equation is an equation that relates an unknown function and its derivatives to some given functions or constants.

The order of a differential equation is defined as the order of the highest derivative appearing in the equation. For example, if a differential equation contains only the first derivative of the unknown function, it is said to be a first-order differential equation. If it contains the second derivative, it is a second-order differential equation, and so on.

The degree of a differential equation is defined as the highest power of the unknown function in the equation. For example, if a differential equation is of the form y' + 2y = 0, the degree of the differential equation is 1 (the highest power of y is 1). If it is of the form y'' + 4y = 0, the degree of the differential equation is 2 (the highest power of y is y^2).

Hence, To find the order and degree of a given differential equation, simply count the highest derivative and the highest power of the unknown function that appear in the equation.

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It will take Jamal's robot 8 seconds to catch up to Kate's robot.

Word problems are sentences describing a 'real-life' situation where a problem needs to be solved by way of a mathematical calculation.

Let's t represent the time it takes for Jamal's robot to catch up to Kate's robot (in seconds).

During this time, Kate's robot will have traveled a distance of 2t + 8 feet.

Jamal's robot will have traveled a distance of 3t feet.

When Jamal's robot catches up to Kate's robot, these two distances will be equal:

2t + 8 = 3t

Solving for t:

3t -2t = 8

t = 8 seconds.

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The absolute error in evaluating sinx is [tex]cos x \triangle x[/tex].

(a)Consider the problem of evaluating the value of sin x.

The objective is to estimate the absolute error in sin x.

Assume r = sin x

The absolute error formula gives that:

[tex]\triangle r = \frac{\delta r}{\delta x} \triangle x[/tex]

[tex]= \frac{\delta sin x}{\delta x} \triangle x[/tex]

[tex]= cos x \triangle x[/tex]

(b) The objective is to estimate the relative error in sin x.

The relative error is given by:

[tex]\frac{\triangle r}{r} = \frac{cos x \triangle x}{sin x}[/tex]

[tex]= cot x \triangle x[/tex]

(c) The objective is to estimate the condition number for this problem.

The formula of a function r = f(x), its condition is given by:

[tex]\frac{x f'(x)}{f(x)}[/tex], here r = sin x

The condition number at point x is:

[tex]\frac{x\frac{d(sin x)}{dx} }{sin x}[/tex]

[tex]= \frac{x cos x}{sin x}[/tex]

= x cot x

(d) The objective is to find for what values of argument x is the problem highly sensitive.

Here, condition number is xcot x.

The cot function is highly sensitive at x=0.

Hence, the argument x is highly sensitive for x=0.

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The value of the sample variance rounded to one decimal place is 76.0.

What is the Mean?

The mean is a measure of the central tendency of a set of data, calculated as the sum of the values divided by the number of values in the set.

The sample variance of a set of data is a measure of the spread or dispersion of the data around the mean. To calculate the sample variance, you can follow these steps:

Calculate the mean of the data: First, find the sum of all the data points and divide by the number of data points (n).

For the given data, the mean is: (-11 + -5 + -5 + -11 + 13 + -11 + -5) / 7 = -7

Subtract the mean from each data point: Next, subtract the mean from each data point to get the deviations from the mean.

For the given data, the deviations are: (-11 - -7) = -4, (-5 - -7) = 2, (-5 - -7) = 2, (-11 - -7) = -4, (13 - -7) = 20, (-11 - -7) = -4, (-5 - -7) = 2

Square the deviations: Square each deviation to get rid of negative values.

For the given data, the squared deviations are: (-4)^2 = 16, (2)^2 = 4, (2)^2 = 4, (-4)^2 = 16, (20)^2 = 400, (-4)^2 = 16, (2)^2 = 4

Sum the squared deviations: Sum all the squared deviations.

For the given data, the sum of the squared deviations is: 16 + 4 + 4 + 16 + 400 + 16 + 4 = 456

Divide the sum by n - 1: Finally, divide the sum of the squared deviations by the number of data points minus 1 (n - 1). This gives us the sample variance.

For the given data, the sample variance is: 456 / (7 - 1) = 456 / 6 = 76

Hence, the value of the sample variance rounded to one decimal place is 76.0.

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The point of concurrency for the lines containing the altitudes of a triangle is called the orthocenter of the triangle.

The orthocenter of a triangle is the point where the perpendicular drawn from the vertices to the opposite sides of the triangle intersect each other.

It is located inside the triangle in an acute triangle. For an obtuse triangle, it lies outside of the triangle. For a right-angled triangle, it lies on the vertex of the right angle.

Therefore, the point of concurrency for the lines containing the altitudes of a triangle is called the orthocenter of the triangle.

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They allowed their opponents to score 134 more runs than they scored.

What is the differential equation?

A differential equation is an equation that contains at least one derivative of an unknown function, either a normal differential equation.

A differential equation is an equation that contains at least one derivative of an unknown function, either a normal differential equation or a partial differential equation.

A run differential of +134 for the Atlanta Braves in the 2021 season means that the team scored 134 more runs than their opponents over the course of the season.

A positive run differential indicates that a team has outscored their opponents, while a negative run differential means that a team has been outscored by their opponents.

Hence, They allowed their opponents to score 134 more runs than they scored.

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Answer:0.69

Step-by-step explanation:

Let z = e^(2x). Then your equation is

 z^2 +3z -28 = 0

 (z +7)(x -4) = 0

 z = -7 or 4

Taking the log of our definition of z, we have

 ln(z) = 2x

 x = ln(z)/2

So, the one real solution is

 x = ln(4)/2 = ln(√4)

 x = ln(2)

The decimal approximation is

 x ≈ 0.69

Answer:

AB≈50.27in²

Step-by-step explanation:

AB=πr2=π·42≈50.26548in²

Answer: 16π squared inches

Step-by-step explanation:

The base area of a cone is the area of a circle.

Formula for area of a circle = π[tex]r^{2}[/tex]

Since your r= 4 inches, you plug in 4 inches into the formula.

Area = π * [tex]4^{2}[/tex] = 16π

Answer:

19

Step-by-step explanation:

step by step

Answer:   x=5.3

Solution:

6x−8x−2=8.6−4x

−2x−2=8.6−4x

2x−2=8.6

2x=10.6

x=5.3

Answer:

The balance after 6 years, assuming the deposit of 5000 earns 4% interest compounded yearly, would be 7400. This is calculated by using the formula A=P(1+r/n)^nt, where A is the balance, P is the principal (5000), r is the interest rate (0.04), n is the number of times the interest is compounded per year (1), and t is the number of years (6).

For the system of equation " 2x₁ – x₂ = h ; -6x₁ + 3x₂ = k " , it will be consistent for all the values of h and k , which satisfy "k = -3h" .

The System Of Linear Equations is considered as consistent if it has at least one solution.

The system of equation is :

⇒ 2x₁ – x₂ = h .....equation(1) ;

⇒ -6x₁ + 3x₂ = k    ....equation(2) ;

Now replacing the equation(2) by its sum with 3 times the first equation, the the system of equations become :

⇒ 2x₁ – x₂ = h .....equation(1) ;

⇒ 0 = k + 3h ;

Simplifying further ,

we have ;

⇒ k = -3h .

Therefore , the system will be consistent for values of "h" and "k" satisfying  k = -3h .

The given question is incomplete , the complete question is

For what values of h and k the system Of Equations is consistent ?

2x₁ – x₂ = h ; -6x₁ + 3x₂ = k

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The augmented matrix for the given  systems of linear equations are obtained. a. x – 3y=5  b. x+2y=0 x+y=1  c. x - y + z=2 x - z=1  d. y + 2x=0 y+z=0

Two matrices are combined to form enhanced matrices.

Thus, augmented matrix for the given  systems of linear equations are obtained as:

a. x – 3y = 5 : [1  - 3 : 5]

b. x+2y=0 x+y=1 : [1  2 : 0]

                            [1   1 :  1]

c. x - y + z=2 and x - z=1  :  [1  -1  1 : 2]

                                           [1  0  -1 :  1]

d. y + 2x=0 y+z=0  : [2  1  0 : 0]

                                [0  1   1 :  0]

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The lengths of all four sides of the tulip garden are equal to the lengths of the corresponding sides of the rose garden, then the two gardens will be congruent.

In order to determine if the tulip garden will be congruent to the rose garden, we must first calculate the lengths of the corresponding sides of each garden. The rose garden's vertices are labeled (x1, y1), (x2, y2), (x3, y3), and (x4, y4). The tulip garden's vertices are labeled (x1, −y1), (x2, −y2), (x3, −y3), and (x4, −y4). The length of the first side of the rose garden can be found using the distance formula, [tex]d = √((x2−x1)^2 + (y2−y1)^2)[/tex]. The corresponding length of the tulip garden can be found using the same distance formula, d = √((x2−x1)^2 + (y2−y1)^2) We can continue this process for the other  three side of both gardens. If the lengths of all four sides of the tulip garden are equal to the lengths of the corresponding sides of the rose garden, then the two gardens will be congruent.

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The definite integral [tex]\int_{10}^{41}[/tex] f(x)dx = 14.

To calculate the definite integral of a function, we can use the Fundamental Theorem of Calculus, which states that the definite integral of a function is equal to the difference between the antiderivative of the function evaluated at the upper and lower limits of the integration interval. In this case, the definite integral is given as [tex]\int_{10}^{41}[/tex] f(x)dx.

Given that F(10) = -4, F(20) = 5, and F(41) = 10, where F(x) is the antiderivative of f(x), we can calculate the definite integral as follows:

[tex]\int_{10}^{41}[/tex] f(x)dx = F(41) - F(10) = 10 - (-4) = 14.

The Fundamental Theorem of Calculus (FTC) is a theorem in mathematics that establishes the connection between differentiation and integration.

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