Find the number of integral solutions of x+y +z = 12, where −3 ≤ x ≤ 4, 2 ≤ y ≤ 11, and
z ≥ 3.

Answer: The answer is 87. when you look at the picture each number is one number apart from each other like look at

86, 87, 88, 89, 90, 91

85 + 1  = 86

86 + 1 = 87

87 + 1 = 88

89 + 1 = 90

90 + 1 = 91

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

y is greater or equal to 0

x is also greater or equal to 0

you can see that only one answer choice can be the answer

the line is x-y = 2

the upper side is x-y <= 2

proves that c is the answer

x-y ≤ 2

y ≥ 0

x ≥ 0 satisfies this graph. Thus option C is correct.

A schematic or visual presentation that organizes the depiction of data or quantities is known as a graph. The relationships connecting two or more objects are frequently represented by the spots on a graph. They are employed to arrange data in order to highlight correlations and patterns. This data is presented as a pattern on a graph.

When y is bigger than 0 and x is also.

The equation is 2x-y.

x-y ≤ 2 on the upper side of the graph is represented in this same way.

x-y ≤ 2; y ≥ 0 and x ≥ 0 will be the correct represented points in the graph. Therefore, option C is the correct option.

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The value of a₃₁ of the arithmetic sequence exists 77.4.

Given: a₅ = 12.4 and a₉ = : 22.4

For the arithmetic sequence a₁, a₂, a₃, ..., the n-th term exists

where d = common difference

a₅ = 12.4,

a₁ + 4d = 12.4 .........(1)

Because a₉ = 22.4,

a₁ + 8d = 22.4 .........(2)

Subtract (1) from (2), we get

a₁ + 8d - (a₁ + 4d) = 22.4 - 12.4

4d = 10

Dividing throughout by 4, we get

d = 2.5

From (1), we get

a₁ = 12.4 - 4 [tex]*[/tex] 2.5 = 2.4

a₃₁ = 2.4 + 30 [tex]*[/tex] 2.5 = 77.4

Therefore, the correct answer is a₃₁ = 77.4

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(a) The solution to the given system of equations is x = 1, y = 2, z = -3

(b) The solutions to the system of equations are (3.4142, -9.2426) and (0.5858, -0.7574)

From the question, we are to solve the given system of equations

The given system of equation is

x + y + z = 0      ----------- (1)

2x + z = -1         ----------- (2)

x - y - z = 2       ----------- (3)

Add equations (1) and (3)

x + y + z = 0      ----------- (1)

x - y - z = 2       ----------- (3)

__________

2x = 2

x = 2/2

x = 1

Substitute the value of x into equation (2) to find z

2x + z = -1        

2(1) + z = -1

2 + z = -1

z = -1 -2

z = -3

Substitute the values of x and z into equation (1) to determine the value of y

x + y + z = 0    

1 + y + -3 = 0

1 + y - 3 = 0

y = 3 -1

y = 2

Hence, the solution to the given system of equations is x = 1, y = 2, z = -3

b.

The given system of equations is

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

6x² - y² - 2y -3 = 0        --------- (2)

From equation (1)

3x + y = 1

y = 1 - 3x -------- (3)

Substitute into equation (2)

6x² - y² - 2y -3 = 0

6x² - (1 -3x)² -2(1 -3x) -3 = 0

6x² - (1 -3x)(1 -3x) -2 + 6x -3 = 0

6x² - (1 -3x -3x +9x²) -2 +6x -3 = 0

6x² - (1 -6x + 9x²) +6x -5 = 0

6x² -1 +6x -9x² +6x -5 = 0

-3x² +12x -6 = 0

3x² -12x +6 = 0

x² -4x + 2 = 0

Solve the quadratic equation by the formula method,

x = [-b±√(b²-4ac)]/2a

a = 1, b = -4 and c = 2

Thus,

x = [-(-4)±√((-4)²-4(1)(2))]/2(1)

x = [4±√(16-8)]/2

x = (4±√8)/2

x = (4+√8)/2 OR (4 - √8)/2

x = 3.4142 OR x = 0.5858

Substitute the values of x into equation (3)

y = 1 - 3x

When x = 3.4142

y = 1 - 3(3.4142)

y = -9.2426

When x = 0.5858

y = 1 - 3(0.5858)

y = -0.7574

Hence, the solutions to the system of equations are (3.4142, -9.2426) and (0.5858, -0.7574)

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We've got a negative slope, but it doesn't look steep enough to go through 6 on the y-axis.

So, it's the last option:

y = - 1.25x + 5.75

Hope this helps!

The side length of the original field is 19 feet.

What is square?

Suppose the original square is x feet wide. Since its the length and width are the same,

So, length of side of enlarged field = x+5

Area of new enlarged field =  side²  = ( x + 5 )²

So, the enlarged shape would be = ( x + 5 ) ( x + 5)

  Then ,  ( x + 5 ) ( x + 5) = 576

               ( x + 5 )²= 576

                x + 5 = √576

               x + 5  = 24

                 x = 24 - 5 ⇒ 19

Hence, The side length of the original field is 19 feet.

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Answer:

x = -1

y = 1

Step-by-step explanation:

Given expressions

1) 2x + 3y - 1 = 0

2) 5x + 2y + 3 = 0

Add 1 on both sides of the 1) equation:

2x + 3y - 1 + 1 = 0 + 1

2x + 3y = 1

Subtract 3 on both sides of the 2) equation:

5x + 2y + 3 - 3 = 0 - 3

5x + 2y = -3

Current System

1) 2x + 3y = 1

2) 5x + 2y = -3

Multiply 2.5 on both sides of the 1) equation

(2.5) 2x + (2.5) 3y = (2.5) 1

5x + 7.5y = 2.5

Current System

1) 5x + 7.5y = 2.5

2) 5x + 2y = -3

Subtract 2) equation from 1) equation

(5x + 7.5y) - (5x + 2y) = 2.5 - (-3)

Get rid of the parenthesis

5x + 7.5y - 5x - 2y = 2.5 + 3

Combine like terms

5x - 5x + 7.5y - 2y = 5.5

0 + 5.5y = 5.5

5.5y = 5.5

Divide 5.5 on both sides

5.5y / 5.5 = 5.5 / 5.5

[tex]\boxed{y=1}[/tex]

Substitute the y-value back to one of the equations to find the x-value

2x + 3y - 1 = 0

2x + 3(1) - 1 = 0

Combine like terms

2x + 3 - 1 = 0

2x + 2 = 0

Subtract 2 on both sides

2x + 2 - 2 = 0 - 2

2x = -2

Divide 2 on both sides

2x / 2 = -2 / 2

[tex]\boxed{x=-1}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

The unit rate of 1/4 book 1/3 hour in books per hour is 3/4 books per hour.

Unit rate is the rate of a quantity in terms of another quantity per unit.

Unit rate(books per hour) = Number of books / Number of hours

= 1/4 ÷ 1/3

= 1/4 × 3/1

= (1×3) / (4×1)

= 3/4 books per hour

Therefore, the unit rate of 1/4 book 1/3 hour in books per hour is 3/4 books per hour.

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The value of x in 6/7x + 1/2 = 7/8 is 7/16

The equation is given as:

6/7x + 1/2 = 7/8

Subtract 1/2 from both sides

6/7x + 1/2 -1/2 = 7/8 -1/2

Evaluate the difference

6/7x = 3/8

Multiply both sides by 7

6x = 21/8

Divides both sides by 6

[tex]x = \frac {7}{16}[/tex]

Hence, the value of x is 7/16

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The correct answers to the sample space illustrated will be:

S = {I, L, L, M, P, T, U, Y}.

S = {g, r, b, y, o}.

S = {W, X, Y, Z}.

Flipping a fair, two-sided coin.

Sample space = lemon-lime, lemon-lime, watermelon, watermelon, watermelon, grape, grape, grape, grape, grape.

S = {Joy, Bob, Ben, Peter, John}

A paper bag has five colored marbles. The marbles are red, green, blue, yellow, and orange. The sample space when choosing one marble will be S = {g, r, b, y, o}.

A four-part spinner is spun once and the spinner with four parts labeled W, X, Y, and Z. The sample space for the experiment is S = {W, X, Y, Z}.

The event will have a sample space of S = {h, t} is the flipping a fair, two-sided coin.

The correct sample space for the gumballs in her bag will be sample space = lemon-lime, lemon-lime, watermelon, watermelon, watermelon, grape, grape, grape, grape, grape.

My experiment with 5 or more possible outcomes will be that a teacher wants to randomly give one of his give students a pack of chocolate.

The sample space for the experiment will be based on their names. This will be S = (Joy, Bob, Ben, Peter, John).

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The situation described by the distance from Town A is the first graph.

Given that cities A,B and C are on straight line.

We are required to find the graph appropriate the given situation.

Distance is basically the measurement of how far two things,places are located.It can be measurement from one point to another .

On the first hour,they leave town B to town C moving farther from Town A,hence the distance is increasing on the first hour.

On the second hour they stop for lunch,meaning that the distance remains constant at the point, it was at the end of the first hour.

Hence if towns A,B C lie on a straight line and Jack and Arthur start at town B and drive for an hour from Town B to Town C then the distance shown from Town A during these two hours is first graph.

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Answer:

The answer is x=2 and y=0.

Step-by-step explanation:

The given Equations are:

                    5x-2y=10      ----------------- (i)
                    4y+20=10x   ----------------- (ii)

From equation find the value of x, that is:

         5x=10+2y         (Obtained By adding 2y on both side of Equation (i))

Next divide with 5 on both side to get x:

          x=(10+2y)/5           ------------------ (iii)

Put this value of x in equation (ii):

        4y+20=10((10+2y)/5)

Simplifying the above equation:

         4y+20=2(10+2y)

         4y+20=20+4y

From the above equation the solution of y is not possible.

We can make it 0. So,
         y=0

Put this value of y in equation (iii), to get value of x:

        x=(10+2(0))/5

        x=(10+0)/5

        x=10/5

        x=2

Given a triangle ABC at points A = (-6, 3) B=(-4,7) C = (-2, 3), and a first transformation of up 2 and right 3, and a second transformation of down 1 and left 6, the final point B" is located at (-7, 8), which is option (a).

In mathematics, a transformation refers to a change or mapping of one set of values to another set. In geometry, a transformation involves changing the position, size, or orientation of a geometric object.

To find the location of the final point B", we need to apply the two transformations given to the initial point B (-4, 7).

First transformation: up 2 and right 3

This means we add 2 to the y-coordinate (to move the point up) and add 3 to the x-coordinate (to move the point right).

So, the new location of B after the first transformation is: (-4 + 3, 7 + 2) = (-1, 9)

Second transformation: down 1 and left 6

This means we subtract 1 from the y-coordinate (to move the point down) and subtract 6 from the x-coordinate (to move the point left).

So, the new location of B" after the second transformation is: (-1 - 6, 9 - 1) = (-7, 8)

Therefore, the final point B" is located at (-7, 8), the correct option is a.

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The probability of getting exactly one face card is 0.72.

According to the questions two cards are drawn from a deck.

Since, we know that

Total number of cards in a deck = 52

Total number of face cards =  3 × 4 = 12

So, the number of ways of selecting 2 cards from a deck = [tex]^{52} C_{2}[/tex]

And, the total number of ways of  getting exactly one face cards

= [tex]^{12} C_{1} \times ^{40} C_{1}[/tex]  + [tex]^{40} C_{1} \times ^{12} C_{1}[/tex]

= 2[tex]^{40} C_{1} \times ^{12} C_{1}[/tex]

Therefore, the probability of getting exactly one face card

= [tex]\frac{2(^{40} C_{1} \times ^{12} C_{1})}{^{52C_{2} } }[/tex]

[tex]= \frac{2(\frac{40 \times 39!}{1!\times 39!} \frac{12\times 11!}{11!}) }{\frac{52\times 51\times 50!}{2!\times50!} }[/tex]

[tex]= \frac{2(40\times 12)}{26\times 51}[/tex]

[tex]=\frac{2\times 480}{1326}[/tex]

= 0.72

Hence, the probability of getting exactly one face card is 0.72.

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It will take 585 billion years to solve the puzzle tower of Hanoi puzzle.

What is hanoi of tower?

The Tower of Hanoi is a mathematical game or puzzle that uses three rods and a number of disks that can slide onto any rod. It is also known as The Problem of the Benares Temple, the Tower of Brahma, Lucas' Tower, or simply the pyramid puzzle. The disks are stacked on one rod at the start of the problem, the smallest at the top and roughly forming a conical shape.

Using the Tower of Hanoi as little as possible

In one version of the problem, 64 golden disks are being used by Brahmin priests to solve it.

Minimum moves required if you had 64 golden disks [tex]=2^{64}-1[/tex]

The problem would take around 585 billion years to solve if every move took a single second.

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The given series is diverges according to the P Test.

According to the statement

we have given that the a series and we have to find that the those series is a converges or diverges.

So, For this purpose, we know that the

P-series test to say whether or not the series converges.

So, The given series is

[tex]1 + 1/\sqrt{32} + 1/\sqrt[3]{243 } + 1/\sqrt[4]{1024} + 1/\sqrt[5]{3125}[/tex]

And then

The value of  P becomes P = 5/6 < 1  when we find it using the p series test.

Hence, the p-series diverges

Because according to the p series test, The series converges if, and only if, the power satisfies p>1.

And diverges if, and only if, the power satisfies p<1.

So, The given series is diverges according to the P Test.

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Question:

Use Theorem 9.11 to determine the convergence or divergence of the p-series. 1 + 1/32 squareroot + 1/243 squareroot 3 + 1/1024 squareroot 4 + 1/3125 squareroot 5.

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Answer:

The coordinates of D is (7, 10)

Explanation:

Let the coordinates of endpoint D be (x, y)

Mid-point formula:

[tex]\sf (x_m, y_m) = (\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2})[/tex]

Applying formula:

[tex]\sf (-1, \:4) = (\dfrac{-9+ x}{2},\: \dfrac{-2+y}{2} )[/tex]

Comparing found:

[tex]\sf -1 =\dfrac{-9+ x}{2} \quad and \quad \: 4 = \dfrac{-2+y}{2}[/tex]

[tex]\sf 2(-1)+9 =x \quad and \quad \: 2(4)+2 =y[/tex]

[tex]\sf x = 7 \quad and \quad \: y = 10[/tex]

So, coordinates of D is (7, 10)

The answer is (7, 10).

Remember the midpoint formula : [tex]\boxed {M = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})}[/tex]

Finding the x coordinate of D

Finding the y coordinate at D

The annual interest paid on the account is $75

The formula for determining annual interest is given as;

Interest = PRT/100

where;

Substitute the values

Annual Interest = 1000 × 7. 5 × 1/ 100

Annual interest = 7500/ 100

Annual interest = $75

Thus, the annual interest paid on the account is $75

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The amount of cash needed to retire baldwin's bond early based on the brokerage will be $5427896.

It should be noted that from the information given, we are to calculate the amount of cash needed to retire baldwin's bond early based on the brokerage.

This will be:

= 6000 + (5756951 × 94.18/100)

= 6000 + 5421896.45

= 5427896

Therefore, the amount of cash needed to retire baldwin's bond early based on the brokerage will be $5427896.

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Magnitude of the momentum of this system just after the collision = 19.2Kgm/s

Law of conservation of momentum states that the total momentum after the collision must be equal to the total momentum before the collision.

The momentum of each ball is given by:

p = mv

where m is the mass of the ball and v its velocity.

The momentum of ball 1 is:

p = mv

= (3.0 kg)(4.0 m/s)

= 12.0 kg m/s in the eastward direction

The momentum of ball 2 is:

p = mv

= (5.0 kg)(3.0 m/s)

= 15.0 kg m/s in the northward direction

The two momenta are in perpendicular directions, so the magnitude of the total momentum can be found as:

[tex]p = \sqrt{p1^{2} +p2^{2} }[/tex]

[tex]p = \sqrt{12^{2} +15^{2} }\\p = \sqrt{144 + 225} \\p = \sqrt{369}\\ p = 19.2 kgm/s[/tex]

and due to the law of conservation of the momentum, this is also equal to the total momentum after the collision.

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The values associated to the exponential function are listed below:

Herein we have the exponential function [tex]g(x) = 1\,296^{x}[/tex], which has to be evaluated, that is, replacing the variable x by constants and make use of algebra properties until result is found.

g(- 1.75) = 3.572 × 10⁻⁶

g(- 1) = 7.716 × 10⁻⁴

g(- 0.25) = 0.167

g(0) = 1

g(0.25) = 6

g(0.50) = 36

g(0.75) = 216

g(1) = 1 296

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The number of combinations of balloons that can be chosen is 1716.

Selections and combinations are synonyms. Combinations represent the choosing of items from a predetermined group of items. We're not trying to arrange anything here. We're going to pick them.

Utilizing the combinations formula, it is simple to determine how many distinct groups of r objects each may be created from the provided n unique objects.

Number of red balloons = 12

Number of blue balloons = 8

First, we figure out how to choose from 13 red and 9 blue balloons.

There are 7 balloons left out of the 29 total (i.e. 29 - 13 - 9)

So, n = 7+7-1

n = 13

r = 7

Number of combinations of balloons = [tex]_{13} C_{7}[/tex]

                                                               = 13!/(13-7)!7!

                                                               = 13!/6!7!

                                                               = 1716

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[tex]\huge\boxed{18\ \text{points}}[/tex]

We'll represent Louise's, Tammy's, Delores's, and Sheryl's point values with [tex]x[/tex], [tex]x+1[/tex], [tex]x+2[/tex], and [tex]x+3[/tex] respectively since each one is 1 point more than the last.

Add all of these values up and set it all equal to [tex]78[/tex].

[tex]x+x+1+x+2+x+3=78[/tex]

Now, simplify.

[tex]4x+6=78[/tex]

Subtract [tex]6[/tex] on both sides.

[tex]\begin{aligned}4x+6-6&=78-6\\4x&=72\end{aligned}[/tex]

Divide both sides by [tex]4[/tex].

[tex]\begin{aligned}\frac{4x}{4}&=\frac{72}{4}\\x&=\boxed{18}\end{aligned}[/tex]

Since Louise's score is [tex]x[/tex], the answer is [tex]18[/tex].

To verify our answer, add the point totals [tex]18[/tex], [tex]19[/tex], [tex]20[/tex], and [tex]21[/tex].

This equals [tex]78[/tex], so we can be sure the answer is correct.

50,400 different anagrams can be made from the letters in the word statistics.

Given word : STATISTICS

Anagrams are words formed by jumbling the positions of the letters given in the word.

For such problems we do factorial of number of words and divide by the repeated letters factorials.

Total number of letters = 10

They can be permutated among themselves in 10! ways.

Some letters are repeated in word STATISTICS

Since, there are 3 S's and 3 T's and 2 I's

So, Number of anagrams = [tex]\frac{10!}{3!.3!.2!}[/tex]

= [tex]\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3!}{3! \times 3 \times 2 \times 1 \times 2 \times 1}[/tex]

= [tex]\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4}{3 \times 2 \times 2}[/tex]

= 10 × 9 × 8 × 7 × 2 × 5

= 50,400

50,400 different anagrams can be made from the letters in the word statistics.

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Answer:

13cm

Step-by-step explanation:

Given the map scale:

1cm on the map : 4km actual distance

4km actual distance : 1cm on the map

1km actual distance : [tex]\frac{1}{4} =0.25cm[/tex] on the map

Therefore,

52km actual distance = 0.25 * 52 = 13cm

Answer: C

Step-by-step explanation:

[tex]\frac{3x+1}{x+8} \div \frac{x-7}{5x}\\\\=\frac{3x+1}{x+8} \times \frac{5x}{x-7}\\\\=\frac{5x(3x+1)}{(x+8)(x-7)}\\\\=\frac{15x^2 +5x}{x^2 +x-56}[/tex]

Answer:

The side length of a square is the square root of its area. We know that:

x2 + 4x + 4 = (x + 2)2

Therefore, the side length of the square is x + 2

The area of a rectangle is the amount of space it would cover on a 2-Dimensional plane. The required value of the width is 25 feet.

A rectangle is a plane shape with equal opposite sides. Its area can be determined by;

Area of a rectangle = length x width

Considering the given question, let the length of the plot be represented by l and its width as w. Given that the width of the plot should be longer than its length by 5, then we have;

w = l + 5

But,

Area of the plot = length x width

500 = l x (l + 5)

       = [tex]l^{2}[/tex] + 5l

This implies that,

500 = [tex]l^{2}[/tex] + 5l

[tex]l^{2}[/tex] + 5l - 500 = 0

Applying the quadratic formula, we have;

l = (-b  ± [tex]\sqrt{b^{2} - 4ac}[/tex]) / 2a

a = 1, b = 5, c = -500

= (-5  ±  [tex]\sqrt{5^{2} - (4*1*-500)}[/tex]/ 2

 = (-5  ± 45) / 2

l =   (40) / 2 OR l =  (-50) / 2

l = 20 OR l = - 25.5

So that,

l = 20

Therefore, the length of the plot is 20 feet.

Thus the width of the plot can be determined as;

w = l + 5

   = 20 + 5

w = 25

The width of the plot is 25 feet.

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Answer: Your answers are

A. 1, 287 ways

B. 154,440 ways

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The point that is a solution to the system of inequalities is (5, 0)

The system of inequalities is given as:

y ≤ 2x + 2

y ≥ -5x + 4

Next, we test the given options.

From the list of options, we have

(x, y) = (5, 0)

Substitute (x, y) = (5, 0) in y ≤ 2x + 2 and y ≥ -5x + 4

y ≤ 2x + 2

0 ≤ 2 * 5 + 2

0 ≤ 12 -- this is true

y ≥ -5x + 4

0 ≥ -5 * 5 + 4

0 ≥ -21 -- this is true

Since both results are true, then it means that the point that is a solution to the system of inequalities is (5, 0)

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Complete question

Which point is a solution to the system of inequalities graphed here? y ≤ 2x + 2

y ≥ -5x + 4

A. (1,6)

B. (-6,0)

C. (0,5)

D. (5,0)