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A function f of three real variables assigns a real number f(x, y, z) to each set of real numbers (x, y, z) in the domain of the function.

Now, According to the question:

Three-Variable Calculus considers functions of three real variables. A function f of three real variables assigns a real number f(x, y, z) to each set of real numbers (x, y, z) in the domain of the function. The domain of a function of three variables is a subset of coordinate 3-space { (x,y,z) | x, y, z ∈ {R} }.

A real-valued function f defined on a subset  D  of  [tex]R^2[/tex]  is a rule that assigns to each point  f(x ,y)  in  D  a real number  f(x ,y) . The largest possible set  D  in  [tex]R^2[/tex]  on which  f  is defined is called the domain of  f , and the range of  f  is the set of all real numbers  f(x ,y)  as  (x ,y)  varies over the domain  D . A similar definition holds for functions  f(x ,y ,z)  defined on points  (x ,y ,z)  in  [tex]R^3[/tex] .

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The solution of the given system of equation is (3, 2).

In the given question we have to find the solution to the system of equations:

y = 4x − 10 and y = 2

We can solve the system of equation using the substitution method.

In substitution method we have to put the the value of one variable from one equation to the other equation.

We have to equations:

y = 4x − 10........................(1)

y = 2...................................(2)

Now putting the value of y from the equation 2 in equation 1

2 = 4x − 10

Add 10 on both side, we get

4x = 12

Divide by 4 on both side, we get

x = 3

Now putting the value of x in equation 1

y = 4(3) − 10

y = 12 − 10

y = 2

Hence, the solution of the given system of equation is (3, 2).

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The right question is:

What is the solution to the system of equations y = 4x − 10 and y = 2?

The value of discriminant of the quadratic equation is 72.

A Discriminant is a function of the coefficients of a polynomial that can be used to determine the number and type of solutions of the polynomial equation. It is often denoted by the letter "D" or "Δ".

It is denoted by this formula

D = b^2-4ac

As per the data given in the questions

We have to calculate the value of the discriminant of the quadratic equation

As we know if the value of discriminant =0 then the value of D=0 and

if the value of discriminant >0 then value of D>0

so, first we have to find D=0 as per the questions.

As we know that the value of D=

Here the value of a=2 , b= 4 and c = -7

D=16+56

D=72

So, the value of discriminant of the quadratic equation is 72.

Hence, the value of discriminant of the quadratic equation is 72.

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Answer:

15/8

Step-by-step explanation:

The ratio is 45:24

45/24 simplify by 3

15/8

So, the ratio in simplest form is 15/8

Prove: <br>

cosec2 A-cos2 A=sec2A -sin2 Atan2A

Answer:

yes

Step-by-step explanation:

Answer:

x = -2.75 and y = 1.5.

Step-by-step explanation:

To solve the system of equations 4x - 2y = -2 and -3x + 5y = -9, we can use the elimination method. This method involves adding or subtracting the equations in a way that eliminates one of the variables, leaving us with an equation in terms of the remaining variable.

First, we can add the equations together to eliminate the y variable:

(4x - 2y) + (-3x + 5y) = (-2) + (-9)

Combining like terms on the left side of the equation gives us:

x + 3y = -11

Next, we can solve for x by dividing both sides of the equation by 4 to get:

x = (-11) / 4 = -2.75

Once we have found the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the second equation, -3x + 5y = -9, and substitute -2.75 for x:

-3(-2.75) + 5y = -9

Solving for y, we get:

y = (-9 - (-8.25)) / 5 = 1.5

Therefore, the solution to the system of equations 4x - 2y = -2 and -3x + 5y = -9 is x = -2.75 and y = 1.5.

Answer:

[tex]y=-3x+7[/tex]

Step-by-step explanation:

[tex]=\frac{4-1}{1-2}=-3 \\ \\ y-4=-3(x-1) \\ \\ y-4=-3x+3 \\ \\ y=3x+7[/tex]

The  limit of a function with a hole is explained below:

For example, consider the function f(x) = 1/x. This function has a hole at x = 0, because it is not defined for x = 0. Therefore, the limit of f(x) as x approaches 0 does not exist.

In all, if a function has a hole at a point x = a, the limit of the function as x approaches a does not exist. However, it is still possible to calculate the limits of the function as x approaches a from the left or the right.

So for example, the limit of f(x) as x approaches 0 from the left (i.e. as x approaches 0 from negative values) is -infinity, and the limit of f(x) as x approaches 0 from the right (i.e. as x approaches 0 from positive values) is infinity.

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Answer:

below

Step-by-step explanation:

x = train A speed

x+5 = train B speed

rate * time = distance

(x     +   x+5 ) * 7 = 665

14x + 35 = 665

14x = 630

x = 45 mi/hr  = A      B = 5+ A = 50 mi/hr

Using trigonometric ratio, the height of the building is 72 feet

The angle of elevation is an angle that is formed between the horizontal line and the line of sight. If the line of sight is upward from the horizontal line, then the angle formed is an angle of elevation.

In this question, we need to use trigonometric ratio to find the height of the building.

Using tangent of the angle;

tan θ = opposite / adjacent

tan 45 = x / 52

x = 52tan45

x = 52

From the top of the building to the window level of the other building, the distance is 52 feet.

The height of the building = 52 + 20 = 72 feet

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The probability of getting 2 six when a fair die is rolled 4 times is given as 1/216.

Probability for any event is the ratio of number of favorable events to the total possible events.

It deals with the measurement of the chance and its value is always in the interval [0, 1].

The sample space and the number of favorable outcomes are given as follows,

The total number of ways two 6 can come out of the 4 times = ⁴C₂ = 6

And, the total number of outcomes for the 4 roll of dice = 6⁴

Now, the probability is given as,

P(6 two times) = 6/6⁴ = 1/216

Hence, the probability of getting 2 six out of 4 throws of a dice is 1/216.

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Answer:

64

Step-by-step explanation:

What is the median of the data 46 64 87?

Median: center most value when values are arranged in numerical order so:

46 , 64 , 87

The roots of the discriminant of the equation 4x² + 3x + 1 = 0 are complex and imaginary.

The discriminant of a quadratic equation is a value calculated by subtracting the square of the coefficient of the x term from four times the coefficient of the x² term, and then multiplying it by the constant term. For this equation, the discriminant is calculated as D = b2 - 4ac, where b = 3 and c = 1. This gives us D = 32 - 4(4)(1) = -7. Since the discriminant is negative, this means that the roots of the equation are complex and imaginary.

D = b2 - 4ac = 32 - 4(4)(1) = -7

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Answer:

m = -3/5

Step-by-step explanation:

The slope is rise/run or (y2 - y1) / (x2 - x1)

We see the y increase by 3 and the x decrease by 5, so the slope is

-3/5

So, m = -3/5

__________________________________________________________

Hello! In this question, we are going to find the slope of the line with the coordinates given to us in the question.

__________________________________________________________

Explanation:

In the question, we were given the points:

In order to find the slope of the line, we will use the slope formula:

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

We will plug our coordinates into the equation and solve.

__________________________________________________________

Solve:

Plug in the coordinates into the equation and solve.

[tex]m=\frac{-3-(-6)}{2-7} \\\\m=\frac{-3+6}{2-7} \\\\m=\frac{3}{2-7} \\\\m=\frac{3}{-5}\\\\\boxed{m=-\frac{3}{5}}[/tex]
Therefore, our slope for the given coordinates would be [tex]-\frac{3}{5}[/tex]

__________________________________________________________

Answer:

[tex]\boxed{m=-\frac{3}{5}}[/tex]

__________________________________________________________

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So X > X inequality have no solution.

What is inequality ?

If an inequality has no real solution, this means that there are no numbers that can be substituted into the inequality to make the statement true. If an inequality has all real numbers as the solution, this means that every real number can be substituted into the inequality to make a true statement.

Have given,

X > X ...........(i)

If suppose the value of X is any amount like 1,2,3,......

put the value of X in equation (i),

1 > 1

1 - 1 > 0 => 0 > 0 That is not possible

Similarly, 2 > 2 => 2 - 2 > 0 => 0 > 0 ,That is not possible

Let X = n

n > n => n - n > 0 => 0 > 0 , it is not possible too

So, X > X inequality have no solution.

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Have given,

X > X ...........(i)

If suppose the value of X is any amount like 1,2,3,......

put the value of X in equation (i),

1 > 1

1 - 1 > 0 => 0 > 0 That is not possible

Similarly, 2 > 2 => 2 - 2 > 0 => 0 > 0 ,That is not possible

Let X = n

n > n => n - n > 0 => 0 > 0 , it is not possible too

So, X > X inequality have no solution.

Answer:

sh. 32

Step-by-step explanation:

Before the reduction, the cost was [tex]35(8/7)=[/tex]sh. 40.

Therefore, the price before the increase was [tex]40(4/5)=[/tex]sh. 32.

The phone's cost that varies directly with time are given as follows:

Both Mark and Nikiya.

The variation of the functions can have two classifications, given as follows:

Mark's cost starts at (0, 20) and goes through (3, 40), meaning that when the input increases from zero to 3, the output increases from 20 to 40, hence it has a direct variation.

Nikiya's cost starts at (0, 0) and goes through (2, 30), meaning that when the input increases from zero to 2, the output increases from 0 to 30, hence it has a direct variation.

The problem asks for whom's function varies directly with time.

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Answer:

Based on the graph shown to the right, whose phone cost varies directly with the amount of time spent talking?  

✔ Nikiya’s only

Mark’s cost does not vary directly with time spent talking because

✔ the cost of 0 hours is not 0 dollars

For Nikiya’s plan, the value of the constant of variation is

15,

which represents Nikiya’s

✔ cost per hour

Step-by-step explanation:

Hope this helps :)

Answer:

  6 inches

Step-by-step explanation:

You want to know the uniform width of border around a 3 ft square coffee table that would have an area of 7 square feet.

The area of the border is x feet wide by 3 feet long along each of the 4 sides, together with 4 corner squares that are x ft square.

  A = 7 = 4(3x +x²)

Completing the square, we have ...

  x² +3x +9/4 = 7/4 +9/4 . . . . . . divide by 4, add (3/2)² to both sides

  (x +3/2)² = 4 . . . . . . . . . . . . simplify

  x +3/2 = √4 = 2 . . . . . . square root

  x = 1/2 . . . . . . . . . foot

The width of the border should be 6 inches.

Answer:

4

Step-by-step explanation:

since there are 7 numbers, divide it by 2. Then you will get 3 remainder 1. Now, count 3 numbers from the first number and the very next number will be your median.

3x + 2y = 5; 2x - 3y = 7. Hence, the given lines are intersecting. So, the given pair of linear equations has exactly one solution and therefore it is consistent.

If a consistent system has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.

If a system has no solution, it is said to be inconsistent . Represent parallel lines.

if Both line slope are same and parallel to each other.

So,  no solution

System is  inconsistent.

if Both equation represent the same line. So, infinite many solution.

System is consistent.

if Both lines are parallel. So, no solution.

System is inconsistent.

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The graph f(x) = (x – 5) The x-axis is touched at -2 by 3(x + 2).

By examining the power of the factor, you may determine whether the graph will cross the x-axis or only touch it.

(x - 5)

3 is an odd number because it has a power of 3. A power of ODD indicates that the graph will pass through the x-axis.

(x + 2)

A power of 2, or even number, exists in the number 2. In the case of an EVEN power, the graph will contact the x-axis.

Set the factor equal to zero and solve to determine the location where it will touch the axis.

Subtract 2 from both sides so that (x + 2) = 0.

x = -2

The complete question is:-

At which root does the graph of f(x) = (x – 5)3(x + 2)2 touch the x-axis?

A. -5

B. -2

C. 2

D. 5

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The surface area of the soccer ball to the nearest square centimeter is 744.68 square centimeters .

Given :

The volume of a soccer ball is 1,930. 75 cubic centimeters .

we know that ,

shape of soccer ball is same as the sphere

Volume of sphere = 4 / 3 * pi * r^3

1930.75 = 4 / 3 * 3.14 * r^3

1930.75 = 4.18 * r^3

r^3 = 1930.75 / 4.18

= 462

r = 7.7 cm

Surface area  of soccer ball = 4 * pi * r^2

= 4 * 3.14 * 7.7^2

= 12.56 * 59.29

= 744.68 square centimeters

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Yes, algebra 3 is definitely a thing.

Algebra is a branch of mathematics that deals with the study of symbols and the rules for manipulating them. It includes the study of mathematical operations and relationships, as well as the properties of equations, terms, and functions. Algebra is a tool for modeling and solving many problems in science, engineering, economics, and other areas.

It is a course designed to build upon the principles of algebra and geometry, and it is typically offered in high school math classes. It covers topics such as linear and quadratic equations, systems of equations, polynomials, matrices, and graphing. Algebra 3 also introduces students to advanced concepts such as logarithmic and exponential functions, probability, and statistics.
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To find the quotient of a polynomial, first divide the polynomial by the divisor. Then use polynomial long division to divide the polynomial by the divisor. The quotient is the result of the polynomial long division.

1. Identify the polynomial to be divided and the divisor.

2. Divide the highest degree term of the polynomial by the divisor.

3. Multiply the divisor by the result from the previous step and subtract this from the polynomial.

4. Bring down the next term in the polynomial.

5. Repeat steps 2-4 until all terms in the polynomial have been divided.

6. The result is the quotient (the answer) of the polynomial division.

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Answer:

Graph A is a function.

Step-by-step explanation:

To be considered a "function", the graph's x-values must have only one corresponding y-value. However, the same y-value can correspond for different x-values. You can determine if there is one x-value for every y-value by using the vertical line test.

. . . . . . .❓ What is the vertical line test?

. . . . . . ✔️ The vertical line test is where you draw vertical lines that       crosses through every x-value on the x-axis. If a line touches more than one point on the function, then the graph is not a function. If a line touches only one point on the function, the graph is a function. You need at least one x-value with points on the function to say that the graph is not a function.

Look at the image I attached to my answer to see the vertical line test used on this problem.

The numbers that are potential roots of the function P(x) = x⁴+22x²-16x-12 are ±6 , ±1 and  ±3 .

The Roots of the equation can be calculated by using the rational roots theorem , which is written as ,

the root x = ± (factors of the constant term)/(factors of leading coefficient).

the function is given as : P(x) = x⁴ + 22x² - 16x - 12

from the above function , we can write that the constant term is = -12 ;

the leading coefficient term is = 1 .

we can write the factors of -12 as ±1, ±2, ±3, ±4, ±6 and ±12 .

the factors of 1 are ±1 .

So , By Rational root theorem , the possible roots are ±1, ±2, ±3, ±4, ±6 and ±12 .

the choices given in the question are ±6 , ±1 , ±3 .

Therefore , the potential roots are (a) ±6  , (b) ±1  and  (c) ±3 .

The given question is incomplete , the complete question is

Which of the numbers below are potential roots of the function P(x)=x⁴+22x²-16x-12  ?

(a) ±6

(b) ±1

(c) ±3

(d) ±8

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A dilation is a type of transformation causes the image to be larger or smaller than the pre-image.

A dilation is a type of transformation that changes the size of a figure without changing its shape. Dilations can make a figure larger or smaller.

In a dilation, the original figure is called the preimage. The dilated figure is called the image. Figures are dilated from a fixed point, called the center of dilation.Dilations also scale all sides of a figure by the same factor. As a result, dilations preserve angle measures and the ratio between corresponding parts.When dilating a figure, the different parts of the figure are multiplied by the scale factor.

In a dilation, the scale factor refers to the ratio between the corresponding sides of the preimage and the image. Therefore, the scale factor, k, can be calculated by dividing the length of one part in the image with the corresponding part in the preimage. The length of the preimage part should always be in the denominator as it belongs to the original figure.

A dilation is a type of transformation causes the image to be larger or smaller than the pre-image.

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The required factor of the given equation 6x²+17x+5 is (3x+1)(2x+5).

Writing a number or other mathematical object as the result of numerous factors—typically smaller or simpler objects of the same kind—is known as factorization or factoring in mathematics.

For instance, the polynomial x² - 4 and the integer 15 can both be factored by the number 3 * 5.

So, we have the equation:

6x²+17x+5

Now, factorize as follows:

6x2+17x+5

6x2+15x+2x+5

3x(2x+5)+1(2x+5)

(3x+1)(2x+5)

Therefore, the required factor of the given equation 6x²+17x+5 is (3x+1)(2x+5).

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Correct question:

Factorise: 6x²+17x+5.