A three-column table is given.


Part A C D
Part 25 35 50
Whole B 56 80


What is the value of B in the table?
44
40
36
30

Answer:

A system of linear equations is a set of two or more equations containing two or more variables. The solution to a system of linear equations is the set of values for the variables that makes all the equations in the system true.

For example, a system of linear equations whose solution is (6,19) could be represented by the following equations:

x + 2y = 25

3x - 4y = 6

Solving this system would yield x = 6 and y = 19 as the solution.

The weight of the coalition P1, P2, P3 is 36

From the question, we have the following parameters that can be used in our computation:

weighted voting system [20: 9,7,5,3,1]

The variable P1, P2 and P3 are the first three values

So, we have

P1 = 20

P2 = 9

P3 = 7

The weight of the coalition P1, P2, P3 is the sum of the points

So, we have

Coalition = 20 + 9 + 7

Coalition = 36

Hence, the weight coalition is 36

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Answer:

Step-by-step explanation:

Answer:

1.91 lb

Step-by-step explanation:

1 3/4 cups of pecans per pie = 1.75 cups.

For 5 pies, 5 × 1.75 cups = 8.75 cups are needed.

1 cup = 99 g

8.75 cup × (99 g) / (1 cup) = 866.25 g

Exact conversion: 453.59237 g = 1 lb

866.25 g × (1 lb)/(453.59237 g) = 1.91 lb

Answer: 1.91 lb

The probability of event a occurring is 0 and the conditional probability of event b occurring given that event a has occurred is 0.5.

The probability that both event A and event B occur is calculated using the formula P(A and B) = P(A) x P(B|A).

In this case, P(A) = 0 and P(B|A) = 0.5. When we plug these values into the equation, P(A and B) = 0 x 0.5 = 0. Therefore, the probability of both event A and event B occurring is 0. The probability that both events a and b occur is determined by multiplying the probability of event a occurring with the conditional probability of event b occurring given that event a has already occurred. This can be expressed mathematically as P(A∩B)=P(A)*P(B|A). In this case, the probability of both events a and b occurring is 0.5*0.5=0.25, since the probability of event a occurring is 0 and the conditional probability of event b occurring given that event a has occurred is 0.5.

This makes sense, since event A has a probability of 0 and cannot occur in the first place, so the probability of both events occurring together is 0.

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A graph that represents the solution set of the compound inequality is: graph D.

In Mathematics, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the following inequality symbols:

Next, we would solve the given compound inequality by making x the subject of formula as follows;

x + 3 < 1/2(4x - 12) < 20

Multiplying all through by 2, we have the following:

2x + 6 < 4x - 12 < 40

Next, we would solve the compound inequality in parts:

2x + 6 < 4x - 12

4x - 2x < 12 + 6

2x < 18

x < 18/2

x < 9.

4x - 12 < 40

4x < 40 + 12

4x < 52

x < 52/4

x < 13.

In conclusion, the line on a number line (graph) should be open when the inequality symbol is greater than (>) or less than (<).

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The conclusion about given functions are given below.

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).

We have the following theorem:

If f is a quadratic function of the form f(x) = ax²+bx+c and ac < 0 , then the function f has two x-intercepts.

(a) g(x) = -8x²+5x-2

For this function a = -8 and c = -2, then -8 × -2 = 16 is greater than zero. Therefore, we cannot conclude anything.

(b) h(x) = [tex]-\frac{1}{3}x^{2} + 3x[/tex]

For this function a = -8 and we don't know the value of c. Therefore, we cannot conclude anything.

(c) k(x) = 8x²-5x-7

For this function a = 8 and c = -7, then 8 × -7 = -56 is less than zero. Therefore, we can conclude that the function k has two x-intercepts.

(d) j(x) = [tex]-\frac{71}{99}x^{2} + 210[/tex]

For this function a = [tex]-\frac{71}{99}[/tex] and c = 210, then [tex]-\frac{71}{99}[/tex] × 210 = [tex]-\frac{4970}{33}[/tex] is less than zero. Therefore, we can conclude that the function k has two x-intercepts.

(e) f(x) = 4x²-3x+7

For this function a = 4 and c = 7, then 4×7 = 28 is greater than zero. Therefore, we cannot conclude anything.

(f) F(x) = [tex]-x^{4} + x^{3} + 9[/tex]

We cannot conclude anything because this is not a quadratic function.

The conclusion about given functions are given above in explanation.

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complete question : Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, a, b, and c are real numbers. Theorem If f is a quadratic function of the form f .x/ D ax2 C bx Cc and ac < 0, then the function f has two x-intercepts. Using only this theorem, what can be concluded about the functions given by the following formulas(a) g (x) = -8x^2 + 5x - 2 (b) h (x) = -(1/3)x^2 + 3x (c) k (x) = 8x^2 - 5x - 7 (d) j (x) = -(71/99)x^2 + 210 (e) f (x) = 4x^2 - 3x + 7 (f) F (x) = -x^4 + x^3 + 9

As per the permutation method, the number of possible way for here travel is 5.

The term permutation in math is known as an arrangement of objects in a definite order and the members or elements of sets are arranged here in a sequence or linear order.

Here we have given that Amanda got a train ticket valid for the five cities listed below.

There are Amsterdam, brussels, Lille, Paris, Rotterdam.

So, the given number of ways is 5.

Then the value of n = 5 and the value of r = 1 because here we have given that she can choose to visit some or none of these cities which means that she must chose at least one city.

Then the number of ways is calculated as,

=> ⁵P₁ =  5!/ (5 - 1)!

=> ⁵P₁ = 5!/4!

=> ⁵P₁ = 5

Complete Question:

Let us consider that Amanda got a train ticket valid for the five cities listed below. she can choose to visit some or none of these cities. she doesn't have time to visit all of them.

They are Amsterdam, brussels, Lille, Paris, Rotterdam

Find out the number of ways to visit each cites using permutation method.

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The direction and position of the fourth and fifth arrows indicate that the object then moves towards the origin, with a smaller velocity.

Motion maps are used to illustrate the direction and position of an object. Using the motion map, the description of the object position and velocity is as follows:

The object starts its movement from the origin with a large velocity, before moving back to the origin with a smaller velocity. It stops for 1 second in the origin, then moves away with a larger velocity, Finally, it moves back towards the origin with a smaller velocity.

See attachment for the motion map, where the number on each arrow in the map, represents the position of the object.

Note that; the long arrow means large velocity while the short arrow means small velocity.

We analyze the direction and position using the arrows

Therefore, the direction and position of the fourth and fifth arrows indicate that the object then moves towards the origin, with a smaller velocity.

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Answer:

Based on the motion map, the object starts at the reference point "x" on the left side of the position line. The three dots with short vector arrows moving toward "x" indicate that the object is moving in the direction of the reference point and gaining speed. This suggests that the object is accelerating. Then, there is a single dot, which might indicate a momentary pause or a constant velocity. Afterward, the two dots with longer vector arrows moving away from "x" suggest that the object is decelerating or slowing down. In summary, the motion map shows an object that starts from rest, accelerates towards the reference point, maintains a constant velocity for a brief moment, and then decelerates away from the reference point.

Step-by-step explanation:

Use chat gpt to give you unique answer.

Answer: 43 because 7 is greater than 5 so you round up. Have a nice day ;)

The correct answer is 43.  0.735 is greater than 0.5, so you round it up to 43

The algebraic expression which represents the total cost of buying watermelons and tomatoes as described is; C = 0.68w + 3.25t.

The cost of buying a watermelon that weighs 18.5 pounds and 5 pounds of tomatoes is; $28.83.

It follows from the task content that the cost of buying Tomatoes are $3.25 / lb and Watermelon is $0.68 / lb.

Hence, the algebraic expression which represents the cost of buying a watermelon that weighs, w pounds and tomatoes that weigh t pounds is;

C = 0.68w + 3.25t.

Hence, if the watermelon bought weighs 18.5 pounds and tomatoes weigh 5 pounds,

C = 0.68 (18.5) + 3.25 (5)

C = 12.58 + 16.25

C = $28.83.

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Answer:

This is nonsense, if GP = 16 and GH = 12 then HP = 4 (16 - 12 = 4). FP is looking basically the same size as HP, so FP = 4, not 2.

Step-by-step explanation:

These are all lies, your teacher is a liar and EF must be around 12 too.

Answer:

30

Step-by-step explanation:

Answer:

The Z-score (also known as a standard score) represents the number of standard deviations a value is away from the mean of a distribution. The formula for the Z-score is:

Z = (X - μ) / σ

Where X is the value of interest, μ is the mean of the distribution, and σ is the standard deviation of the distribution.

Given the information in the problem, we know that the mean of the fruit fly's lifespan is 32 days and the standard deviation is 5 days.

To find the Z-score of a fruit fly whose lifespan is 37 days:

Z = (37 - 32) / 5 = 1

To find the Z-score of a fruit fly whose lifespan is 30 days:

Z = (30 - 32) / 5 = -0.4

To find the Z-score of a fruit fly whose lifespan is 47 days:

Z = (47 - 32) / 5 = 2.4

Therefore, a fruit fly whose lifespan is 37 days has a Z-score of 1, a fruit fly whose lifespan is 30 days has a Z-score of -0.4, and a fruit fly whose lifespan is 47 days has a Z-score of 2.4.

The remaining portions from the three pizzas together make up 13/27 of a large pizza.

A fraction is defined as a numerical representation of a part of a whole that represents a rational number.

Shauntee ate 2/3 of her pizza, so she has 1/3 left.

Carlos ate 8/9 of his pizza, so he has 1/9 left.

Rocco ate 26/27 of his pizza, so he has 1/27 left.

To determine the combined fraction of the remaining portions of the three pizzas, we add the fractions that are left:

1/3 + 1/9 + 1/27 = (1/3) + (1/9) + (1/27)

Simplifying this fraction by finding the least common denominator, which is 27:

(1/3) + (1/9) + (1/27) = (9/27) + (3/27) + (1/27) = 13/27

So they together make up 13/27 of a large pizza.

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The complete table is given as follows -

Brain (part)    0.6    1.3    1.8     3

Body (part)    6.6   14.3  19.8   33

Total (whole) 7.2   15.6  21.6   36

Given is that an interior designer recently spent £1359.55 buying 25 kettles and 20 toasters for a smaller group of new houses. For her current assignment she needs 80 kettles and 56 toasters.

The given table is -

Brain (part)   0.6   B   1.8      E

Body (part)   A    14.3   19.8  33

Total (whole) 7.2  C      D     36

We can write the expressions as -

0.6 + A = 7.2                                           .... Eq( 1 )

A = 6.6

1.8 + 19.8 = D                                            .... Eq( 2 )

21.6 = D

E + 33 = 36

E = 3                                                              .... Eq( 3 )

Therefore, the complete table is given as follows -

Brain (part)    0.6    1.3    1.8     3

Body (part)    6.6   14.3  19.8   33

Total (whole) 7.2   15.6  21.6   36

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It is verified that the given vectors x₁ and x₂ are solutions of the given system. Using Wronskian it is shown that they are linearly independent. Then, [tex]W(t)=7e^{-3t}[/tex]. The general solution of the given system is written as [tex]x(t)=\left[\begin{array}{c}3c_1e^{2t}+c_2e^{-5t}\\2c_1e^{2t}+3c_2e^{-5t}\end{array}\right][/tex].

Wronskian analysis helps to determine whether a solution is linearly dependent or independent.

Given the system is [tex]x'=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x[/tex].

Let's differentiate x₁ concerning t, and we get,

[tex]\begin{aligned}x_1'&=\left[\begin{array}{c}\frac{d}{dt}(3e^{2t})&\\\frac{d}{dt}(2e^{2t})&\end{array}\right] \\&=\left[\begin{array}{c}6e^{2t}&\\4e^{2t}&\end{array}\right] \end{aligned}[/tex]

Now,

[tex]\begin{aligned}\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x_1&=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]\left[\begin{array}{c}3e^{2t}\\2e^{2t}\end{array}\right]\\&=\left[\begin{array}{c}12e^{2t}-6e^{2t}\\18e^{2t}-14e^{2t}\end{array}\right]\\&=\left[\begin{array}{c}6e^{2t}\\4e^{2t}\end{array}\right]\\&=x_1'\end{aligned}[/tex]

From this, we can write,

[tex]x_1'=\left[\begin{array}{cc}4&-3\\6&-7\end{array}\right]x_1[/tex]

Therefore, we conclude that x₁ is a solution to the given system.

Let's differentiate x₂ concerning t, and we get,

[tex]\begin{aligned}x_2'&=\left[\begin{array}{c}\frac{d}{dt}(e^{-5t})&\\\frac{d}{dt}(3e^{-5t})&\end{array}\right] \\&=\left[\begin{array}{c}-5e^{-5t}&\\-15e^{-5t}&\end{array}\right] \end{aligned}[/tex]

Now,

[tex]\begin{aligned}\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x_2&=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]\left[\begin{array}{c}e^{-5t}\\3e^{-5t}\end{array}\right]\\&=\left[\begin{array}{c}4e^{-5t}-9e^{-5t}\\6e^{-5t}-21e^{-5t}\end{array}\right]\\&=\left[\begin{array}{c}-5e^{-5t}\\-15e^{-5t}\end{array}\right]\\&=x_2'\end{aligned}[/tex]

From this, we can write,

[tex]x_2'=\left[\begin{array}{cc}4&-3\\6&-7\end{array}\right]x_2[/tex]

Therefore, we conclude that x₂ is also a solution to the given system.

Now, find the Wronskian of x₁ and x₂, we get,

[tex]\begin{aligned}W(t)&=\text{det}[x_1\;x_2]\\&=\text{det}\left[\begin{array}{cc}3e^{2t}&e^{-5t}\\2e^{2t}&3e^{-5t}\end{array}\right] \\&=(3e^{2t}\times3e^{-5t})-(e^{-5t}\times2e^{2t})\\&=9e^{2t-5t}-2e^{2t-5t}\\&=9e^{-3t}-2e^{-3t}\\&=7e^{-3t}\\&\neq0\end{aligned}[/tex]

From this, we can conclude that x₁ and x₂ are independent.

Finally, we write the general solution of the system as follows,

[tex]\begin{aligned}x(t)&=c_1x_1+c_2x_2\\&=c_1 \left[\begin{array}{c}3e^{2t}\\2e^{2t}\end{array}\right] +c_2\left[\begin{array}{c}e^{-5t}\\3e^{-5t}\end{array}\right] \\&=\left[\begin{array}{c}3c_1e^{2t}\\2c_1e^{2t}\end{array}\right] +\left[\begin{array}{c}c_2e^{-5t}\\3c_2e^{-5t}\end{array}\right]\\&=\left[\begin{array}{c}3c_1e^{2t}+c_2e^{-5t}\\2c_1e^{2t}+3c_2e^{-5t}\end{array}\right] \end{aligned}[/tex]

The complete question is -

First, verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system.

[tex]x'=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x;\; x_1=\left[\begin{array}{ccc}3e^{2t}\\2e^{2t}\end{array}\right], \;x_2=\left[\begin{array}{cc}e^{-5t}\\3e^{-5t}\end{array}\right][/tex]

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When we are aligning two shafts, the reason behind rotating the coupling hubs before taking each measurement is so that each of the measurements is taken from the same place on the hub every time.

Shafts are basically a component of circular cross section which rotates and transmits the power from a driving device like an engine or a motor through a particular machine. Coupling hubs are basically the attachment of the coupling to shaft in a device.

During the alignment of two shafts, the angular as well as parallel misalignment is thoroughly checked. The coupling hubs are rotated before taking each measurement is so that each of the measurements is taken from the same place on the hub every time.

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The inequality equation is 8.498 > 8.478

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the inequality equation be represented as A

Now , the value of A is

Let the first number be represented as = p

Let the second number be represented as = q

Now , the value of p = 8.498

The value of q = 8.478

And , the hundredths value in the number 8.498 is greater than the hundredths value in the number 8.478

So , the value of p is greater than q

Therefore , 8.498 > 8.478

Hence , the inequality is 8.498 > 8.478

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In a case whereby you mix 1 half cup of salt with 3 cups of sugar the ratio of salt to sugar.

The concept that will be used is ratio. An ordered pair of numbers a and b, represented as a / b, is a ratio if b is not equal to 0. A proportion is an equation that sets two ratios at the same value. For instance, you could express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls)

we were told that there were 1 half cup of salt with 3 cups of sugar

Then the ratio can be written as

salt/sugar

= (1 1/2)/3

Since the amount of the salt is in mixed fraction, we can reduce to improper fraction for easy calculation.

=( 3/2)/ 3

= 3/6

=1/2

Hence, the ratio is 1:2

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missing options:

A. 2:3

B.1:2

C.2:1

D. 5:6

The expected waiting time in hours until the first call arrived is 0.00045.

The formula for poisson distribution is expressed as

P(x = r) = (e^- µ × µ^r)/r!

Where ,

µ=  represents the mean of the theoretical distribution.

r = represents the number of successes of the event.

According to the question,

µ = 5            

For the probability that there are one in one hour, it is expressed as

P(x = 1) ,

Therefore,

P(x = 1)

= (e^- 10 × 10^1)/1! = 0.00045

Therefore,

P(x = 1)

= 0.00045

Poisson distribution

A Poisson distribution is a discrete probability distribution. It gives the probability of an event happening a certain number of times (k) within a given interval of time or space.

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The numeric value of the function is given as follows:

2x + h + 9.

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

The function in this problem is given as follows:

f(x) = x² + 9x.

At x = x + h, the function is defined as follows:

f(x + h) = (x + h)² + 9(x + h)

f(x + h) = x² + 2xh + h² + 9x + 9h

f(x + h) = x² + 2xh + h² + 9x + 9h.

Subtracting by f(x), we have that:

f(x + h) - f(x) = h² + 2xh + 9h.

f(x + h) - f(x) = h(h + 2x + 9).

Dividing by h, the numeric value is given as follows:

2x + h + 9.

Meaning that the first option is the correct option.

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Answer:

3,2 °C

Step-by-step explanation:

40,2 °C - 37 °C = 3,2 °C

we can conclude that option (A) , (B) and (C) are true.

We know that in a triangle,

• Sum of any to sides is greater than the third side.

• Difference between any two sides is smaller than the third side.

So, in ∆EFG, we have,

• EF + FG > EG

• FG + EG> EF

• EF + EG > FG

and,

• EF - FG <EG or FG - EF <EG

• FG - EG < EF or EG - FG < EF

• EF - EG <FG or EG - EF < FG

From options now we get,

A) EF + FG > EG is true

B)EG + FG > EF is true

C) EG - FG < EF is true

D) EF - FG > EG is false

E) EG + EF < FG is false

Therefore, we can conclude that option () , () and () are true.

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We can conclude that option (A) , (B) and (C) are true.

We know that in a triangle,

• Sum of any to sides is greater than the third side.

• Difference between any two sides is smaller than the third side.

So, in ∆EFG, we have,

• EF + FG > EG

• FG + EG> EF

• EF + EG > FG

and,

• EF - FG <EG or FG - EF <EG

• FG - EG < EF or EG - FG < EF

• EF - EG <FG or EG - EF < FG

From options now we get,

A) EF + FG > EG is true

B)EG + FG > EF is true

C) EG - FG < EF is true

D) EF - FG > EG is false

E) EG + EF < FG is false

Therefore, we can conclude option (A) , (B) and (C) are true.

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The factors and zeros of the function f(x)=x2-5x+6 are x-3, x-2 and 3 and 2 respectively

Given the quadratic function as;

f(x) = x² - 5x + 6

Using the factorization method of solving quadratic function, we have to multiply the coefficient of x² by the constant of the function

Then find the pair factors of the function that sums up to -5

The pair factors area -2 and -3

Substitute the values, we get;

x²- 2x - 3x + 6

Pair the expression

(x² -2x) - (3x + 6)

Factorize the value

x(x -2) -3(x - 2)

The zeros are;

x-3 = 0

x = 3

Then,

x -2 = 0

x =2

Hence, the values are 2 and 3

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Internet of Things refers to the technology that allows data, collected from sensors in all types of machines, to be sent over the Internet to repositories where it can be stored and analyzed.

Internet of Things refers to the collective network of connected devices and the technology that facilitates communication between devices and the cloud, as well as between the devices themselves.

There are some Advantages of IOT:

It can assist in the smarter control of homes and cities via mobile phones. It enhances security and offers personal protection.

Automating activities saves us a lot of time. Information is easily accessible, even if we are far away from our actual location, and it is updated frequently in real time.

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Answer:

  6√2

Step-by-step explanation:

You want one side of a square whose half-diagonal has length 6.

If X is the center point of the square where the diagonals cross at right angles, triangle EXG is an isosceles right triangle with leg length 6. The hypotenuse of an isosceles right triangle is √2 times the leg length, so ...

  EG = 6√2

__

Additional comment

If you like, you can use the Pythagorean theorem to prove that.

  EG² + EX² +GX² = 6² +6² = 72

  EG = √72 = √(36·2) = 6√2

A square is a rhombus, so the diagonals cross at right angles. A square is a rectangle, so the diagonals are the same length. A square is a parallelogram, so the diagonals bisect each other.

The presented system of linear equations has no solution.

The final form of the augmented matrix suggests that the system of linear equations has no solution, as the last row represents an equation with 0 = 0, which is always true but doesn't provide any information about the variables.

To see this more clearly, let's use the augmented matrix to write out the system of linear equations:

1x - 1y + 2z = 4

2y + 1z = -1

z = 1

The first two equations are consistent and have a unique solution, but the last equation is trivial and provides no information about x and y. In other words, infinitely many pairs of x and y satisfy the first two equations, but none satisfy the third equation. Therefore, the given system of linear equations has no solution.

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In the horizontal direction and by 4 units to the left, f(x) should be shifted to match g(x).

The correct option is A.

The function provided by the basic function f(x) and the constant

g(x) = f(x - k),

may be drawn by horizontally moving the f(x) k unit coordinates.

The shift's direction depends on the value of k. Specifically,

The base graph moves k units to the right if k > 0, and

The base graph moves k units to the left if k < 0.

Given:

The quadratic functions f(x) and g(x) are described in the table.

From the table, the functions,

f(x) = x².

And g(x) = (x + 4)².

In the attached image,

the blue curve is the function of f(x).

And red line is the function of g(x).

From the graph,

Left by 4 units, f(x) should be shifted  to match g(x).

Therefore, left by 4 units, f(x) should be shifted to match g(x).

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Answer: A

Step-by-step explanation: took the test

Both f(x) and g(x) have one x-intercept at x = -1/3.

Theorem A states that for a cubic function of the form f(x) = x^3 + x + b, with b > 1, the function f has exactly one x-intercept. Using Theorem B, we can conclude that for any nonzero real number k, the function g(x) = kf(x) = kx^3 + kx + bk also has exactly one x-intercept.

For example, if f(x) = x^3 + x + 5 and k = 2, then g(x) = 2x^3 + 2x + 10. Both f(x) and g(x) have exactly one x-intercept. This can be shown by solving the quadratic equation formed by setting f(x) = 0 or g(x) = 0. Doing this yields x = -1/3. Therefore, both f(x) and g(x) have one x-intercept at x = -1/3.4

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The population of the town at any given time t can be determined using the formulas provided. For example, if the town's population at time t=5 years is desired, the population can be calculated as follows:(a) P(5) = 1600 + 70(5) = 2100 people (b) P(5) = 1600(1.09)^5 = 2266.05 people

(a) P(t) = 1600 + 70t

(b) P(t) = 1600(1.09)^t

(a) In this case, the population increases by 70 people per year, so the formula for the population as a function of time is P(t) = 1600 + 70t. This means that the population is 1600 people at time t=0 and increases by 70 people for each additional year.

(b) In this case, the population increases by 9 percent each year, so the formula for the population as a function of time is P(t) = 1600(1.09)^t. This means that the population is 1600 people at time t=0 and increases by a factor of 1.09 (9 percent) for each additional year.

The population of the town at any given time t can be determined using the formulas provided. For example, if the town's population at time t=5 years is desired, the population can be calculated as follows:

(a) P(5) = 1600 + 70(5) = 2100 people

(b) P(5) = 1600(1.09)^5 = 2266.05 people

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