I showed a picture of the question

Yes, [tex]T(x_1,x_2,x_3)=(x_1+x_3,2x_2-x_3)[/tex] is a linear transformation.

A linear map is a mapping V to W between two vector spaces in mathematics, more specifically in linear algebra, that keeps the operations of vector addition and scalar multiplication. The more broader example of modules over a ring also uses the same terminology and definition are Module homomorphism.

The maps between algebraic objects are known as homomorphisms. Group homomorphisms and ring homomorphisms are the two basic categories. In Other words ,homomorphisms of modules, algebras, and vector space homomorphisms, which are also known as linear maps.

Two properties must be satisfied  if  T is a linear transformation:

T(x+ y)  = T(x) + T(y) for any x, y in [tex]R^3.[/tex]

homogeneity is T(ax) = a T(x). For all scalars a and x in [tex]R^3.[/tex].

let a consider

L.H.S,T(x+y)= T[tex]((x_1+y_1),(x_2+y_2),(x_3+y_3[/tex]))= [tex](x_1+y_1+x_3+y_3, 2x_2+2y_2-x_3)[/tex]

R.H.S. T(x) + T(y) =[tex](x_1+x_3, 2x_2-x_3) +(y_1+y_3, 2y_2-y_3) =(x_1+y_1+x_3+y_3, 2x_2+2y_2-y_3).[/tex]

L.H.S.=R.H.S.

So,The first property is satisfied i.e. T(x+ y)  = T(x) + T(y) for any x, y in [tex]R^3.[/tex]

For homogeneity,

let a consider ,

L.H.S. T(ax) = T[tex](ax_1,ax_2,ax_3) = (ax_1+ax_3, 2ax_2-ax_3).[/tex]

R.H.S. a T(x) = a T[tex](x_1,x_2,x_3) = a(x_1+x_3, 2x_2-x_3) = (ax_1+ax_3, 2ax_2-ax_3).[/tex]

L.H.S.=R.H.S.

So, T(ax) = a T(x). for all scalars a and all vectors x in  [tex]R^3.[/tex].are satisfied

A linear transformation from [tex]R^3[/tex]to[tex]R^2[/tex]  . is therefore represented by T.

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Ian earned $500 in simple annual interest after 10 years, while Jackson earned $1,128.89 more in interest than Ian after 10 years due to compound interest.

Ian's account earns simple annual interest, which means that he earns 5% interest on his initial deposit every year. After 10 years, he will have earned

interest = principal x rate x time

interest = 1000 x 0.05 x 10

interest = $500

Jackson's account earns interest that is compounded annually, which means that he earns 5% interest on his initial deposit plus any interest that has already been earned every year. After 10 years, he will have earned

[tex]balance = principal x (1 + rate)^{time}[/tex]

balance = 1000 x (1 + 0.05)¹⁰

balance = $1,628.89

To find out how much more interest Jackson earned than Ian, we need to subtract the interest Ian earned from the balance in Jackson's account

$1,628.89 - $500 = $1,128.89

Therefore, Jackson earned $1,128.89 more in interest than Ian after 10 years.

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Answer:

To find the factored form of the polynomial function, we start by using the zeros to write out the factors of the polynomial. If the zeros are a, b, c, d, then the factors are (x-a), (x-b), (x-c), and (x-d).

In this case, the zeros are -1, -3, 3, and 1, so the factors are (x+1), (x+3), (x-3), and (x-1). Multiplying these factors together gives the polynomial function:

f(x) = (x+1)(x+3)(x-3)(x-1)

To simplify this expression, we can use the distributive property to expand the factors:

f(x) = (x+1)(x^2 - 9)(x-1)

Next, we can multiply the remaining factors using the distributive property and combining like terms:

f(x) = (x^3 - 8x - 9)(x-1)

Finally, we can use the distributive property again to expand the remaining factor:

f(x) = x^4 - 9x^2 + 8x + 9x^3 - 8x - 9

Simplifying this expression gives the factored form of the polynomial function:

f(x) = (x+1)(x+3)(x-3)(x-1) = x^4 - 9x^2 + 8x - 9x^3 - 8x - 9.

Therefore , the solution of the given problem of volume comes out to be 4500 pounds of the crushed mineral can fit inside the container.

The volume of a three-dimensional item, which is measured in cubic units, describes how much room it occupies. Litre and in3 are the symbols for cubic measurements. However, in order to determine an object's dimensions, you must be aware of its volume. It's common practise to translate an object's weight onto metric units like kilogrammes.

Here,

We must first determine the container's volume, then multiply it by the crushed mineral's weight per cubic foot.

The container's measurements in the image are as follows:

=> 5 feet in length

=> Size: 3 feet wide

=> Height is two feet.

The following formula determines the container's volume:

=> Volume = length* width*height

Inserting the values:

=> 5 feet * 3 feet * 2 feet = volume.

=> 30 cubic feet is the volume.

=> Volume of container x Weight per cubic foot = Weight of crushed mineral

=>  Weight of crushed mineral = 30 cubic feet x 150 pounds/cubic foot

=>  The crushed mineral weighs 4500 pounds.

Therefore, 4500 pounds of the crushed mineral can fit inside the container.

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Answer:

False

Step-by-step explanation:

(x+4)^2 =  x^2 + 8x + 16

The volume of the fish tank is 408in³

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.

The volume of regular object is given as;

V = base area × height

base area = 13 × 6

= 78in²

The area of the cut out section = 1/2(a+b) h

= 1/2 × (3+6) × 6

= 9× 3 = 27in²

The actual area of the base shown = 78-27

= 51 in²

therefore the volume of the tank will be

V = 51 × 8

= 408in³

therefore the volume of the tank is 408in³

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Answer:

Step-by-step explanation:

Hopefully this is right.

Area = r²

Which is pi times the the radius squared

The radius is 2, so 2² is 4, So we multiply 3.14, which is pi, by 4.

Multiply 3.14 by 4, and the answer is 12.56, the question says round to the nearest tenth, so the final answer will be 12.6

My math teacher back in 7th grade gave us this sentence to help us memorize the equations.

Cherry Pie Delicious (C=d), Apple Pie Are Too (A=r²)

Do the same with the other two questions, and the answer for question two will be 3.14, rounded to the nearest tenth will be 3.1

Hope this helps!

For a circle with a radius of 2, the area can be calculated using the formula:

Area = π x r^2

where π is approximately 3.14 and r is the radius.

Plugging in the values, we get:

Area = 3.14 x 2^2

Area = 3.14 x 4

Area = 12.56

Rounding to the nearest tenth, we get:

Area ≈ 12.6

Therefore, the area of the circle with radius 2 is approximately 12.6.

For a circle with a radius of 1, the area can be calculated using the same formula:

Area = π x r^2

Plugging in the values, we get:

Area = 3.14 x 1^2

Area = 3.14

Rounding to the nearest tenth, we get:

Area ≈ 3.1

Therefore, the area of the circle with radius 1 is approximately 3.1.

The total amount that Meihua pays for the 36 pencils is  9M - 36n

We know that a set of 4 pencils has a price M, then the price of each single pencil is:

P = M/4

And we also know that Meihua got a discount of n in each pencil, so the price that she pays for each pencil is:

P = M/4 - n

Now we know that she buys 36 of these, so the amount that she pays is given by the product:

amount  = 36*(M/4 - n)

amount = 9M - 36n

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A graph of the data set is shown in the image attached below.

In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.

This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).

In this scenario and exercise, we would use an online graphing calculator to graphically represent the given data set on a number line as shown in the image attached below.

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Answer: $702.39

Step-by-step explanation: To find the total cost of the laptop with sales tax included, we need to add the sales tax to the listed price.

First, we need to calculate the amount of sales tax.

Sales tax = 9.75% of 639.99

Sales tax = 0.0975 x 639.99

Sales tax = 62.40

Now we can find the total cost of the laptop with sales tax included:

Total cost = listed price + sales tax

Total cost = 639.99 + 62.40

Total cost = 702.39

Therefore, the total cost of the laptop with sales tax included is $702.39.

Answer:

A) 150%

B) 20.67 J

Step-by-step explanation:

A) To determine the efficiency of the engine, we can use the formula:

efficiency = (work output / heat input) x 100%

In this case, the engine does 62 J of work each cycle, so the work output is 62 J. We're also given that the engine absorbs 2 times as much heat as it discharges to the environment, so we can express the heat input as 2 times the heat discharged:

heat input = 2 x heat discharged

Therefore, we can rewrite the efficiency formula as:

efficiency = (62 J / (2 x heat discharged)) x 100%

Simplifying this expression, we get:

efficiency = 31 J / heat discharged

Now we need to determine the value of heat discharged. We know that the engine absorbs 2 times as much heat as it discharges, so the net heat input is 3 times the heat discharged:

net heat input = heat input - heat discharged = 2 x heat discharged - heat discharged = heat discharged

Therefore, we can express the net heat input as:

net heat input = 3 x heat discharged

Now we can use the first law of thermodynamics, which states that the net heat input is equal to the work output plus the heat discharged:

net heat input = work output + heat discharged

Substituting the given values, we get:

3 x heat discharged = 62 J + heat discharged

Solving for heat discharged, we get:

heat discharged = 20.67 J

Now we can calculate the efficiency:

efficiency = 31 J / 20.67 J = 1.50 or 150%

Therefore, the efficiency of the engine is 150%.

B) To determine the quantity of heat discharged to the environment each cycle in Joules, we can use the value we calculated above for heat discharged:

heat discharged = 20.67 J

Therefore, the quantity of heat discharged to the environment each cycle is 20.67 J.

Hope this helps!

The collection of all points in a plane that are equally spaced from a given point, referred to as the circle's center, is referred to as a circle. The radius of the circle is the distance measured from any point on the circle to its center.

A circle is described as the collection of all points on a plane that is at the same distance, or radius, from a fixed point known as the circle's center.

Every location inside the circle is, therefore, closer to the center point than the radius, and any point outside the circle is farther away from the center point than the radius.

Therefore, a circle is made up of points that are spaced out by a particular amount, known as the radius, from the circle's center.

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The value of CE is 14.7 units.

The diameter of a given circle is a straight line which passes through the center of the circle and divides it into two equal semi-circles.

So that in the given circle with center A, we have;

CE = BD  (diameter of a given circle)

BA = EA = CA = DA    (radius of a given circle)

CE is the diameter of the circle, and BA is the radius, so that;

CE = 2*BA

     = 2*7.35

     = 14.7

CE = 14.7

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Answer:  formula 9x - 5y = -160 is not a direct variation.

Step-by-step explanation: The provided expression, 9x - 5y = -160, does not exemplify direct variation.

Direct variation is a correlation existing between two variables, wherein one is a constant scalar multiple of the other. Stated differently, should one variable increase by a defined magnitude, the corresponding variable will also experience a commensurate increase of identical magnitude.

The formula provided indicates the absence of a constant proportionality between the temperature in degrees Celsius (x) and the temperature in degrees Fahrenheit (y). It is noteworthy that the correlation amidst the variables x and y is non-linear in nature, thereby precluding its representation in the form of a direct proportionality.

Answer: It is not a direct variation.

Step-by-step explanation:

         While a linear equation is a direct variation, the y-intercept of a direct variation relationship must be 0. Here, that is not the case. Let us manipulate this equation into a slope-intercept equation.

Given:

   9x - 5y = -160

Add 5y and 160 to both sides of the equation:

   5y = 9x + 160

Divide both sides of the equation by 5:

   y = [tex]\frac{9}{5}[/tex]x + 32

32 ≠ 0

The justification of the  angles with a combined degree measure of 90⁰ are complementary angles is proved by part 3. ∠1 is complementary to ∠2.

In the figure,

Measure of angle 1 = 40 degrees.

Measure of angle 2 = 50 degrees

Angle 2 is complementary to angle 3.

Proof of the result,

Measure of ∠1 = 40 degrees  ( given  )

Measure of ∠2 = 50 degrees ( given )

Measure of ∠1 + measure of ∠2 = 40° + 50°

⇒ Measure of ∠1 + measure of ∠2 = 90°

⇒ Angle 1 and angle 2 are complementary to each other.

Complementary angles are such pairs of angles whose sum is equal to 90°.

It is given that

Angle 2 is complementary to angle 3.

This implies ,

m ∠1 + m ∠2 = m ∠2 + m ∠3

⇒ m ∠1  = m ∠3

⇒ m ∠1  ≅ m ∠3

Therefore, the part 3 . ∠1 is complementary to ∠2 gives the justification for the proof of complementary angles.

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Answer:

Brother asked a very good question.

We can write the given arithmetic expressions into equations as follows:

70 + 15 = 85

This can be written as an equation: x + 15 = 85, where x is the unknown value that needs to be determined.

125 + 85 = 210

This can be written as an equation: y + 85 = 210, where y is the unknown value that needs to be determined.

These equations can be used to solve for the unknown values x and y by performing inverse operations (such as subtraction or division) to isolate the unknown variable on one side of the equation.

Answer:

To find the surface area of a rectangular prism, we can use the formula:

Surface Area = 2lw + 2lh + 2wh

where l, w, and h are the length, width, and height of the rectangular prism, respectively.

Given that the rectangular prism is 12 units high, 5 units wide, and 10 units long, we can substitute these values into the formula:

Surface Area = 2(10)(5) + 2(10)(12) + 2(5)(12)

Surface Area = 100 + 240 + 120

Surface Area = 460 square units

Therefore, the surface area of the rectangular prism is 460 square units

Answer:

Yes, it is a polynomial identity (work below).

Step-by-step explanation:

Let's start by expanding the left side of the equation.

[tex](2x^2+y^2)^2=\\4x^4+y^4+4x^2y^2[/tex]

Now, let's expand the second side.

[tex](2x^2-y^2)^2+(2xy\sqrt{2} )^2=\\4x^4+y^4-4x^2y^2+4x^2y^2(2)=\\4x^4+y^4-4x^2y^2+8x^2y^2[/tex]

There are no like terms on the left side, so let's combine the like terms on the right side.

[tex]4x^4+y^4-4x^2y^2+8x^2y^2=\\4x^4+y^4+4x^2y^2[/tex]

Now, let's check if the left and right sides are equal:

[tex]4x^4+y^4+4x^2y^2= 4x^2+y^4+4x^2y^2[/tex]

Both sides are the same. Thus, the equation is a polynomial identity.

Answer is down below!

Step-by-step explanation:

5 x2 =`5 xj2

The probability that the pink can was chosen because the cookie was a sugar cookie is about 0.282, or about 28.2%.

Probability is simply how likely something is to happen. When we are uncertain about the outcome of an event, we can talk about the probability of certain outcomes - how likely they are. The analysis of events driven by probabilities is called statistics.  

We can use Bayes' theorem to calculate the probability that the pink can was chosen because the cookie was a sugar cookie. Let P(P) be the probability that the pink jar was chosen, P(S) be the probability that the chosen cookie  was a sugar cookie, and P(S|P) be the probability that a sugar cookie was chosen given that the pink one was chosen. Then we have:

P(P|S) = P(S|P) * P(P) / P(S)

We can calculate all these probabilities as follows:

P(S|P) = 9 / (15 9 13) = 9/37

This is the probability of choosing a sugar cookie, given that the pink can was chosen.  

P(P) = 1/4

This is the probability of choosing the pink can.

To calculate P(S), we need to consider all possible options for choosing a sugar cookie. We can choose a sugar loaf from a blue, green  or  pink jar. So:

P(S) = P(S|B) * P(B) P(S|G) * P(G) P(S|P) * P(P)

where P(S|B) is the probability of choosing a sugar cookie from the blue jar, P(G) is the probability of choosing the green jar, and so on. We can calculate all these probabilities as follows:

P(S|B) = 12 / (7 12) = 12/19

P(S|G) = 8 / (6 8 5) = 8/19

P(S|P) = 9 / (15 9 13) = 9/37

P(B) = 1/2

P(G) = 1/4

P(P) = 1/4

So,

P(S) = (12/19) * (1/2) (8/19) * (1/4) (9/37) * (1/4) = 271/444

We can now calculate P(P|S) using Bayes' theorem:

P(P|S) = (9/37) * (1/4) / (271/444) ≈ 0.282

Therefore, the probability that the pink can was chosen because the cookie was a sugar cookie is about 0.282, or about 28.2%.

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The calculated value of the perimeter of triangle ABD is 48 units.

Given that

A(1, 1), B(17, 1) and D(17, 13)

To find the perimeter of triangle ABD, we need to find the lengths of its three sides and add them up.

We can use the distance formula to find the lengths of the sides.

The distance between points A and B is:

AB = √((17 - 1)^2 + (1 - 1)^2) = √(256) = 16

The distance between points B and D is:

BD = √((17 - 17)^2 + (13 - 1)^2) = √(144) = 12

The distance between points D and A is:

DA = √((1 - 17)^2 + (1 - 13)^2) = √(256 + 144) = √(400) = 20

Therefore, the perimeter of triangle ABD is:

AB + BD + DA = 16 + 12 + 20 = 48

So, the perimeter of triangle ABD is 48 units.

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Answer:

We know that the x-coordinate of point A is -11, and that the distance between point A and point B is 15 units. Let's call the y-coordinate of point A "y". Then we can use the distance formula to find the two possible values of y:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1,y1) = (-11,y) is the coordinates of point A, and (x2,y2) = (1,2) is the coordinates of point B. Substituting the values, we get:

15 = sqrt((-11 - 1)^2 + (y - 2)^2)

15 = sqrt(144 + (y - 2)^2)

15^2 = 144 + (y - 2)^2

225 = 144 + (y - 2)^2

81 = (y - 2)^2

y - 2 = ±9

y = 2 ± 9

So the possible coordinates of point A are (-11,11) and (-11,-7).

Option 3. x+y≤6 , 32x + 20y ≥ 150. The inequality 32x + 20y ≥ 150 ensures that they bake enough cookies to meet the requirement of making at least 150 cookies.

The right disparity that addresses what is happening is:

32x + 20y ≥ 150

To see the reason why, we want to consider the quantity of treats Luka and Dana can heat in every hour and the all out number of hours they need to prepare. We should expect they have H hours to prepare, where H = 6 for this situation.

Luka and Dana can prepare 32 chocolate chip treats in 60 minutes, so they can heat a sum of 32x treats in x hours. Likewise, they can prepare a sum of 20y snickerdoodle treats in y hours. To make no less than 150 treats, they need to prepare a mix of chocolate chip and snickerdoodle treats in every hour, with the end goal that the complete number of treats they make in H hours is something like 150.

Thus, the imbalance 32x + 20y ≥ 150 guarantees that they heat an adequate number of treats to meet the necessity of making somewhere around 150 treats.

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The missing values of h(x) to show it is a linear function are:

X=1, h(x)=3

X=3, h(x)=17

X=5, h(X)=31

So, the correct answer is (c)

A linear function is a mathematical relationship between two variables that produces a straight line when plotted on a graph, with a constant rate of change between them.

A function is a mathematical rule that assigns each input value to a unique output value. It can be represented graphically, algebraically, or with a table of values.

According to the given information:

To prove that h(x) is a linear function, we need to show that the difference in h(x) values between any two points is proportional to the difference in their corresponding x values.

Using the given values in the table, we can calculate the differences in h(x) values and x values:

Δh(x) = h(x) - h(x-1)

Δx = x - (x-1) = 1

Δh(x=1) = h(x=1) - h(x=0) = ? - (-4) = ?

Δh(x=3) = h(x=3) - h(x=2) = ? - 10 = ?

Δh(x=5) = h(x=5) - h(x=4) = ? - 24 = ?

If h(x) is a linear function, then the ratio of Δh(x) to Δx should be the same for all pairs of consecutive x values. We can use the given values of Δh(x=2) and Δx=1 to calculate this ratio:

Δh(x=2)/Δx = (10 - (-4))/2 = 7

We can now use this ratio to find the missing values of h(x):

Δh(x=1)/Δx = ?/1 = 7

Δh(x=3)/Δx = ?/1 = 7

Δh(x=5)/Δx = ?/1 = 7

?/1 = 7  =>  ? = 7

Therefore, the missing values of h(x) are:

X=1, h(x)=3

X=3, h(x)=17

X=5, h(X)=31

So, the correct answer is (c)

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It can be found by graphing the points given and sketching the parabola. Another point on the graph is (-1, 9) that passes through them. You can move over and up from the vertex and do the same on the opposite side of the line x = -1.

In Mathematics and Geometry, the vertex form of a quadratic function is represented by the following mathematical equation:

f(x) = a(x - h)² + k

Where:

Based on the information provided, the vertex is located at point (2, 0), and this ultimately implies that the axis is represented by the vertical line x = 2 that passes through the vertex.

Additionally, the parabola would be bilaterally symmetric about the above axis, and the mirror image of the point (5, 9) would also be on this parabola.

Therefore, the mirror image would have the same y-coordinate of 9 while the x-coordinate is given by:

x = 2 - (5 -2)

x = -1.

In conclusion, the required point is (-1, 9).

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Complete Question:

The graph of a quadratic function has a vertex (2, 0). One point on the graph is (5, 9). Explain how to find another point on the graph. It can be found by graphing the points given and sketching the parabola. Another point on the graph is (-1, _) that passes through them. You can move over and up from the vertex and do the same on the opposite side of the line x=

Step-by-step explanation:

x^2 -6x - 72   =  ( x-12)(x+6)

If you can't do this 'in your head'  you can always use Quadratic Formula

  with  a = 1      b= - 6    and   c= -72

      to find roots   x = 12 and  -6      then write the equation as above