2. How do you know from the equation that a line is vertical or horizontal?

Answer: y = -1/2x + 4.

The equation of line passes through the points (0, 4) and (8, 0) will be;

⇒ y = -1/2x + 4

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

Two points on the line are (0, 4) and (8, 0).

Now,

Since, The equation of line passes through the points (0, 4) and (8, 0)

So, We need to find the slope of the line.

Hence, Slope of the line is,

m = (y₂ - y₁) / (x₂ - x₁)

m = (0 - 4) / (8 - 0)

m = - 4 / 8

m = -1/2

We know that, The slopes of parallel lines are equal.

Thus, The equation of line with slope -1/2 is,

⇒ y - 4 = -1/2 (x - 0)

⇒ y - 4 = -1/2x

⇒ y = -1/2x + 4

Therefore, The equation of line passes through the points (0, 4) and

(8, 0) will be;

⇒ y = -1/2x + 4

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. To find the distance between two points, use the distance formula.

It will be the square root of (x2-x1)squared + (y2-y1)squared.

Use (3,-4) as x1, y1. Use (5,1) as x2, y2.

We get the square root of (5-3)squared + (1- -4)squared.

That gives us the square root of 2 squared + 5 squared.

Now we have the square root of 4 + 25, = the square root of 29.

That is the distance between the two points.

The perimeter of the rectangle is, 3y + 1 feet.

What is perimeter?

The circumference of a shape is known as its perimeter. The lengths of all four sides must be added in order to determine a rectangle's or square's perimeter. In this instance, the rectangle's length is represented by x and its width by y.

P = x + x + y + y represents the perimeter, P.

Let the length of triangle is 7.5y + 8 feet and the width of triangle is 6y - 7.5 feet.

So, the perimeter of rectangle is,

Perimeter = 2(length + width)

                = 2(7.5y + 8 + 6y - 7.5)

                 = 2(1.5y + 0.5)

Perimeter = 3y + 1

Hence, the perimeter of the rectangle is, 3y + 1 feet.

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Computing on demand, on the cloud, and in real time IT Enviroment is most impacted by Are is known as gird computing.

What is Data Analysis?

Data analysis is the methodical application of logical and/or statistical approaches to describe and demonstrate, summarise and assess, and assess data.

Concept:

Step 1: A network of connected computers that act as a single supercomputer to complete complex tasks like modelling or data analysis is known as gird computing.

Step 2 : On-demand computing is a delivery method in which users are provided with computing resources as needed.

Step 3 : Real-time computing refers to a computer system that responds to an event by completing a task in a predetermined amount of time. a system for air traffic management as an example.

Grid computing, on-demand computing, and real-time computing

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Answer: D; 360

Step-by-step explanation: To find percentages, I like to set up proportional relationships to find the amount. It would look like this:

[tex]\frac{90}{100}[/tex] = [tex]\frac{?}{400}[/tex]

90=  part %

100= whole %

?= amount of the %

400= all of the item(s)

So, we would cross multiply the 90 and the 400, then divide it by 100. That would be 36,000. So, that divided by 100 leaves us with 360. I hope this helped!

The matrix equation Ax=b with one unique solution says that column A is linearly independent.

Two vectors are linearly dependent if at least one of them can be expressed as a linear combination of the others. If no vector can be expressed as a linear combination of the others, then two or more vectors are said to be linearly independent.

Given the matrix equation is Ax=b. And this equation has one unique solution. Then, let us assume this unique solution is x₁.  Let the column values of A be A₁, A₂, A₃, .... Aₙ.

If there is only one solution to the equation Ax = b for all b, then assume b = 0 and come to the conclusion that there is only one solution to the equation Ax = 0 as well. Consequently, the only solution is straightforward, and the columns of A are linearly independent.

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Hi, To convert a temperature from degrees Fahrenheit to degrees Celsius, you can use the following formula:

C = (F - 32) / 1.8

Substituting the value of F and solving for C, we find that:

C = (284 - 32) / 1.8

= 252 / 1.8

= 140

Therefore, 284 °F is equivalent to 140 °C.

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The total amount paid during the lease period is £45,665.64.

To find the total amount paid during the lease period, we need to add the initial payment of £10,000 to the present value of the four equal payments, which can be calculated using the present value formula:

PV = FV / (1 + r)^n

Where PV is the present value, FV is the future value, r is the discount rate, and n is the number of periods. In this case, the present value of the four equal payments is £26,496, the discount rate is 8%, and the number of periods is 4. We can solve for the future value by rearranging the formula:

FV = PV × (1 + r)^n

Plugging in the given values, we get:

FV = £26,496 × (1 + 0.08)^4

FV = £26,496 × 1.34

FV = £35,664.64

This is the total future value of the four payments. To find the total amount paid during the lease period, we need to add this to the initial payment of £10,000. Therefore, the total amount paid during the lease period is:

£35,664.64 + £10,000 = £45,665.64

Therefore the total amount paid during the lease period is £45,665.64.

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Answer: 60 miles per 4 gallons. If that's the question.

Step-by-step explanation:

The possible formula for P(x) is x⁵-x⁴-21x³+45x²

What is a polynomial?

Polynomial is formed composed of the phrases Nominal, which means "terms," and Poly, which means "many." An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable). Based on the number of terms present in the expression, it is categorised as monomial, binomial, and trinomial.

P(x) has roots of multiplicity 2 at x=3 and x=0.

The equation stands, P(x) = x²(x-3)²

P(x) has a root of multiplicity 1 at x=-5

The equation stands, P(x) = x²(x-3)²(x+5)

Simplifying the equation,

P(x) = x²(x²+9-6x)(x+5)

⇒ P(x) = (x⁴+9x²-6x³)(x+5)

⇒ P(x) = x⁵+5x⁴-6x⁴-30x³+9x³+45x²

⇒ P(x) = x⁵-x⁴-21x³+45x²

The possible formula for P(x) is x⁵-x⁴-21x³+45x²

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The critical values for the function are t = 6 , -5.667 where t = 6 makes sense . At t = 6 , y is increasing

According to the question,

After t hours of production, a plant produces an average of y units per hour, according to analysis of its daily output is

y = 102t + 0.5t² - t³ , 0 ≤ t ≤ 10

(a) To calculate critical values for the given function

We have to differentiate it w.r.t time and then have to put y'=0

=> y = 102t + 0.5t² - t³

Derivating w.r.t time

=> y' = 102 + t - 3t²

Equating to zero,

=>  102 + t - 3t² = 0

Solving quadratic equation,

=> t = 6 , -5.667 are critical values

(b) t = 6 makes sense in particular problem because time cannot be negative.

(c) To check increasing value

Derivate y' w.r.t time again

=> y" = 1 - 6t

Putting critical value,

At t = 6

y" < 0 , Therefore at t = 6 function is increasing

Hence , For t = 6 y is increasing

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Derivative of function is 5(2x + 4) * (x² + 4x + 6)⁴.

What is differentiation?

In arithmetic, the derivative of a function of a true variable measures the sensitivity modification of the perform price with relation to a change in its argument. Derivatives are a elementary tool of calculus.

Main body:

by using product rule;

Let = u = x² + 4x + 6

⇒ du/dx = 2x + 4 .

Now y = u⁵

⇒ dy/dx = 5u⁴ .

dy/dx = dy/dx * du/dx = 5u⁴ * (2x + 4)

= 5 * (x² + 4x + 6)⁴ * (2x + 4)

= 5(2x + 4) * (x² + 4x + 6)⁴

hence , derivative of function is 5(2x + 4) * (x² + 4x + 6)⁴.

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The missing value of x = 6

What is value ?

Value in mathematics can refer to a number of closely related ideas. A mathematical value can generally be any specific mathematical object. This is most frequently a number in elementary mathematics, such as a real number like or an integer like 42.

Depending on where it is in the number, each digit has a different value, which is referred to as value. We figure it out by multiplying the digit's place value by its face value. Place value plus face value equals value.

Values are the benchmarks or ideals by which we judge the acts, traits, possessions, or circumstances of others. Values that are embraced by many include those of beauty, honesty, justice, peace, and charity.

4200 / x = 700

If the fraction is defined, the denominator cannot equal 0:

x ≠ 0

Append the inequality as the domain:

x ≠ 0

Multiply both sides of the equation by the common denominator:

400x / x = 700 x

Reduce the fractions:

4200 = 700 x

Swap the sides:

700 x = 4200

Divide both sides of the equation by the coefficient of variable:

x = 4200 / 700

Cross out the common factor:

x = 6

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Answer:

1/2

Step-by-step explanation:

(5/3-7/4)+√1/16+√1/9

= {(5x4-7x3)/12} + 1/4 + 1/3

= -1/12 + (3+4)/12

= (-1+7)/12

= 6/12

= 1/2

Answer:

125+25n

Step-by-step explanation:

We are given that

Heather opens a bank account with=$125

Each week she deposited money  into her account=$25

We have to find the explicit rule for the amount of money that has Heather after n weeks.

Heather has money after 1 week=125+25=$150

Heather has money after 2 week=125+2(25)=$175

Heather has money after 3 weeks=125+3(25)=$200

Heather has money after n weeks=125+25(n)

Hence, the explicit rule for the amount of money that has Heather after n weeks is given by

125+25n

Step-by-step explanation:

Area of circle 1= pie r²

=3.14 x 5²

=3.14x 25

= 78.5km²

Area of circle 2=3.14 x 5.2²

= 3.14 x 27.04

= 84.9km²

change in area= 84.9-78.5

=6.4

radius²= 6.4/ 3.14

r²=2.03

r =√2.03

r=1.42

The four integers say a,b,c and d. If a and b are congruent to moulo m and c ,d are congruent to modulo m then ac and bd are must be congruent to modulo m .

Two integers a and b are "congruent modulo n" if their difference is a multiple of n or in other words we say n/(b-a) . For example, 17 and 5 are congruent modulo 3 because 17 - 5 = 12 = 4×3

We have, a ≡ b mod m => m/(a - b)

and c ≡ d mod m => m/(c-d)

Now, a + c - b - d = (a-b) + (c-d)

because, m/( a - b) and m/(c-d)

=> m/( a-b+ c-d )

=> m/(a+c)-(b+d)

=> a+c ≡ b+d mod m

We have to prove that ac ≡ bd mod m.

ac - bd = c(a-b) + b(c- d)

Now, as we see, m/(a-b) => m/c(a-b)

and m/(c-d) => m/b(c-d)

=> m/(c(a-b) + b(c-d))

=> m/(ac - bc + bc - bd)

=> m/(ac - bd)

=> ac ≡ bd mod m

Hence proved.

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The function for the value of the Porsche after t years is [tex]y = 845000(1-\frac{32}{100} )^t[/tex].

Function is a combination of different types of variable and constants in which for the different values of x the value of function y is unique.

Given that,

The initial price of the 918 Spyder Prosche car = $845,000

Since, the price of car depreciated at the rate of 32% per year.

The function for the value of Porsche car, after t years, can be given as,

The price of car after t year is y,

[tex]y = 845000(1-\frac{32}{100} )^t[/tex]

The required function, the models, the value of the Porsche after t years is [tex]y = 845000(1-\frac{32}{100} )^t[/tex].

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Observing the provided equation, We can't solve these equations using substitution method or any other methods because these lines are parallel and never intersect each other.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.

The algebraic approach to solving simultaneous linear equations is known as substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies.

x=4-8y

3x+24=12

Taking first equation and rearranging it we get,

x+8y=4

multiply this with 3, we get 3x+24y=12 which are identical in nature. Which shows these two lines are parallel in nature. We can't solve them because parallel lines never intersect.

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Observing the provided equation, We can't solve these equations using substitution method or any other methods because these lines are parallel and never intersect each other.

What do you mean by equations?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.

What is a substitution method?

The algebraic approach to solving simultaneous linear equations is known as substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies.

x=4-8y

3x+24=12

Taking first equation and rearranging it we get,

x+8y=4

multiply this with 3, we get 3x+24y=12 which are identical in nature. Which shows these two lines are parallel in nature. We can't solve them because parallel lines never intersect.

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The sum will be an odd integer.

What is an integer?

An integer  is an integer (not a fraction) that is positive, negative, or zero.

In mathematics, integers are collections of integers and negative numbers. Like whole numbers, whole numbers do not contain fractions. Therefore, we can say that integers are numbers that can be positive, negative, or zero, but which are not fractions.

Let p,q,r be three random numbers

Let S be the sum of two even integers and one odd integer

S = 2p + 2q + (2r + 1)

S = 2p + 2q + 2r + 1

S = 2(p+q+r)+1

This shows that S is odd integer for all values ​​of p,q,r

Therefore, the sum will be an odd integer.

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Answer: percent: 25% amount: 40 base: unknown

Base after finding it: 160

Step-by-step explanation:

40/x = 25%/100

butterfly method

25x/25 = 4000/25

simplify; cancel out

x (aka the base) = 160

The required equivalent expression is given as 5(w - 3).

The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.

Here,
An equivalent expression of any expression is given as the simplifying the expression using various operations such as factors.
So,
Given expression,
=  5w - 15
As in the expression 5 is a common term between 5w and 15.
Take 5 commons and put the rest of the expression in the parenthesis.
= 5(w - 3)

Thus, the required expression is 5(w - 3).

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Answer:

21/79

Step-by-step explanation:

(7x6)÷2[{45÷3(7×2-15+6)}+4)​

(7*6)/2[{15(7*2-15+6)}+4)

(42)/2[{15(14-15+6)}+4)

(42)/2[{15(5))}+4)

(42)/2[{75}+4)

(42)/2[75 + 4]

(42)/2 * 79

42/2 * 79

21/79

The total cost of painting and carpeting the 4 walls and the floor is C = ₹(720x² + 1324.8x).

Given is that a room three times as long as it is broad and its height is 4.6 m. The cost of carpeting its floor at Rs. 240 per square meter be  the cost of painting the 4 walls at Rs. 24 per sq. m.​

Assume that the breadth to be equivalent to [x] meters. Then its length will be [3x] meters.

Area of floor = {3x} x {x} = 3x² sq. m.​

The cost of carpeting the floor = Rs. 240 per sq. m.

For 1 sq. m., the cost is 240 sq. m.

For {3x² sq. m}, the cost will be -

C{f} = 240 x 3x² = 720x²

C{f} = 720x²

The total area of the 4 walls = 4 × (3x) × 4.6 = 55.2x sq. m.

The cost of carpeting the walls = Rs. 24 per sq. m.

For {55.2x sq. m}, the cost will be -

C{w} = 24 x 55.2x = 1324.8x

C{w} = 1324.8x

The total cost of painting and carpeting -

C = C{w} + C{f} = 720x² + 1324.8x

C = ₹(720x² + 1324.8x)

Therefore, the total cost of painting and carpeting the 4 walls and the floor is C = ₹(720x² + 1324.8x).

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8 and 7 are missing frequency in the incomplete distribution.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

Class Interval                       frequency            Cumulative frequency                  

0 -   10                                         3                                3                

10 - 20                                         a                              3+a                

20 - 30                                       20                           23+a              

30 - 40                                        12                           35+a              

40 - 50                                         b                          35+a+b  

Mode = 26    hence lies in 20 - 30

f of modal class = 20

Mode  = 20  +  10( 20 - a) / (20 - a + 20 - 12)

26 = 20 + 10 (20 - a) / (28 - a)

60(28 - a)  = 10 ( 20 - a)

168 -6a = 200 - 10a

4a = 32

a = 8

Class Interval                       frequency            Cumulative frequency                  

0 -   10                                         3                                3                

10 - 20                                         8                                11                

20 - 30                                       20                              31              

30 - 40                                        12                              43            

40 - 50                                         b                            43+b  

Median = 27

27  =  20  + 10  ( (43 + b)/2 - 11)/ 20

7 =  ( (21 +  b)/ 4

28 =     (21 +  b)

b = 7

Hence, the missing frequencies are 8 and 7.

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Correct option will be (C) In the short run, firms may incur economic losses or earn economic profits, but in the long run they earn normal profits.

The short run is a concept that states that, within a certain period in the future, at least one input is fixed while others are variable. In economics, it expresses the idea that an economy behaves differently depending on the length of time it has to react to certain stimuli. The short run does not refer to a specific duration of time but rather is unique to the firm, industry or economic variable being studied.

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Your chances of being killed or injured in such a crash are 5 times greater when you are thrown from your car than when you are buckled up

The safest decision that drivers and passengers can make is to use their seatbelts.

The national seat belt use rate was 90.4% in 2021, indicating that many Americans are aware of the life-saving benefits of seat belts.

In 2017, it is projected that 14,955 lives were saved by seat belt use in passenger automobiles.

Recognize the potentially devastating repercussions of not using a seat belt and discover how to always buckle up properly for you and your family.

Buckling up keeps you safe and secure within your car, whereas not doing so increases your chance of being completely ejected from the vehicle in an accident, which is virtually always fatal.

You need more protection than air bags.

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If the price is reduced to $11.40 the number of DVDs sold will be 1177.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that a video store manager observed that the number of DVDs sold seems to vary inversely to the price per DVD. If the store sells 710 DVDs per week when the price per DVD is $18.90.

The number of DVDs sold for $11.40 will be calculated by forming the inverse equation for the number of DVDs.

N = K / P

Here, N is the number and P is the price and K is the inverse constant.

The value of K will be,

K = N x P

K = 18.90 x 710

K = 13419

The number of DVDs for $11.40 will be,

N = K / P

N = 13419 / 11.40

N = 1177

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C(n) = min(αn, βn) then gcd(0,b) = p(1)^(c1)∙p(2)^(c2)..................p(n)^(cn). So for all other a, b have gcd exist.

The integer a=0 and b=0 does not have any greatest common divisor.

Suppose n = gcd(0,r)

Then 12|0 and 12|0 also n is the greatest divides both (a=0), (b=0).

But n+1|0, n+1|0 as (n+1)*0 = 0 which is a contradiction to n = gcd(0, r).

Hence, gcd(0, r) does not exist.

But if a≠0 or b≠0.

It has a unique prime factorization that is;

a= p(1)^(α1)∙p(2)^(α2)∙p(3)^(α3)......................p(n)^(αn),

b= p(1)^(β1)∙p(2)^(β2)∙p(3)^(β3)......................p(n)^(βn).

Let C(n) = min(αn, βn) then gcd(0,b) = p(1)^(c1)∙p(2)^(c2)..................p(n)^(cn).

So for all other a, b have gcd exist.

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