What is set and example?

The distance from point [tex]$a\left(-1,\ 7\right)$[/tex] to the line [tex]$y=3x$[/tex] is about 4.5 units, rounded to the  nearest tenth.

The distance from point [tex]$a\left(-1,\ 7\right)$[/tex] to the line [tex]$y=3x$[/tex] is equal to the length of the perpendicular line from point [tex]$a$[/tex] to the line[tex]$y=3x$[/tex]. The formula to find the distance from a point [tex]($x_1, y_1$)[/tex] to a line [tex]$ax+by+c=0$[/tex] is given by:

[tex]$$\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}.$$[/tex]

Substituting[tex]$$\frac{|ax_1+by_1+c|}[/tex] into the formula, we get:

[tex]$$\frac{|-3(-1)+1(7)+0|}{\sqrt{3^2+1^2}} = \frac{20}{\sqrt{10}}.$$[/tex]

Therefore, the distance from point [tex]$a\left(-1,\ 7\right)$[/tex] to the line [tex]$y=3x$[/tex] is about 4.5 units, rounded to the nearest tenth

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The nature of the roots of a quadratic equation is described by the discriminant calculated by seeing the equation and comparing it with the standard equation which is given by -

[tex]ax^{2} + bx+c[/tex] , where a , b , c are real numbers .

The root of the equation is given by -

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

Here, the term (b^2 - 4ac) is called the discriminant of the quadratic equation .

This discriminant is the defining factor of the nature of the roots.

Now, here arise three cases-

(i) If (b^2 - 4ac) > 0, the roots are real and distinct .

(ii) If (b^2 - 4ac) = 0 , the roots are real and equal.

(iii) If (b^2 - 4ac) < 0 , the roots are not real and they are imaginary i.e. they are complex in nature and of the form a+ib.

Hence, it is the discriminant which determines the nature of roots.

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Answer:

  2. negative infinity to five thirds and five thirds to infinity

Step-by-step explanation:

You want the domain of the rational function f(x) = (x +4)/(3x -5).

The domain is the set of input values for which the function is defined. A rational function is undefined when its denominator is zero. f(x) is undefined when ...

  3x -5 = 0

  x = 5/3

So, the value 5/3 must be excluded from the domain of f(x). The domain of f(x) is ...

  -∞ < x < 5/3  ∪  5/3 < x < ∞

Answer:

3 units.

Step-by-step explanation:

In the equation x = 1/12 (y-1)², the vertex of the parabola is at the point (1,1). The focus of the parabola is located a distance called the "focal length" from the vertex.

The focal length is given by the formula f = 1/(4*p), where p is the coefficient of the x term in the standard form of the equation of the parabola.

In this case, the coefficient of the x term is 1/12, so the focal length is f = 1/(4*(1/12)) = 3. This means that the distance from the focus to the vertex is 3 units.

Answer:

Step-by-step explanation:

Square's mission statement is simple. Square deliver “Economic Empowerment” to the thousands of business owners who use Square to manage their businesses.

The image of (-4, 5) after a translation right 5 units and 2 units up is (1, 7).

What is the image of the point?

The given point P is "reflected" in the mirror and appears on the other side of the line an equal distance it. Drag the point P to see this. The reflection of the point P over the line is by convention named P' (pronounced "P prime") and is called the "image" of point P.

Going right means the x value goes up. In this case, it went to the right 5 spaces. so, all you have to do is add 5 with -4 which is 1.

When going up on a graph, the y value goes up as well. Add 2 with 5 which is 7.

Therefore, the answer is  (1, 7).

Hence , the image point (-4, 5) after a translation right 5 units and 2 units up is (1, 7).

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Yes, because the 16 feet ladder will reach to the window that is 14.5 feet above the ground.

Pythagoras' Theorem states that the square of the hypotenuse side of a right-angled triangle equals the sum of the squares of the other two sides.

The hypotenuse of a triangle, which has a straight angle of 90 degrees, has a square that, according to the Pythagoras theorem, equals the sum of the squares of the other two sides.

According to the question;

A 20-foot ladder is leaning against the side of a house.

Hypotenuse H= 20 feet

The bottom of the ladder is 12 feet from the base of the house.

Base B = 12 feet

And let P be the perpendicular.

Using Pythagoras theorem

H²= P² + B²

20² =12² +P²

P²= 400-144

P²= 256

Taking the square root.

P= 16 feet.

Since the height of the ladder is 16 feet, and we have to reach 14.5 feet above the ground.

16 > 14.5

That means, the ladder will reach to the window.

Hence, Yes, the 16 feet ladder will reach to the window.

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Answer:

B. (w)(y)

Step-by-step explanation:

(w)(y)

= 2 * √8 * 3 * √2

= 2 * √(2*2*2) * 3 *√2

= 2 * √2 * √2 * √2 * 3 * √2

= (2*3) * (√2 * √2) * (√2 * √2)

= 6 * 2 * 2

= 24

The linear function rule to model the distance d in feet of the car traveled after t seconds is d = 83t.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

The distance traveled after 9 seconds = 747 ft.

The function rule to model the distance d after t seconds.

d = 83t

When t = 9,

d = 83 x 9 = 747 ft

Thus,

The linear function is d = 83t.

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The factors of 6x² 5x 6 are: 1, 2, 3, 5, 6, 10, 15, 30, x, 2x, 3x, 5x, 6x, 10x, 15x, 30x, x², 2x², 3x², 5x², 6x².

1. Find the common factors of 6x², 5x, and 6:

The common factors are 1, 2, 3, 5, and 6.

2. Find the factors of each individual term:

For 6x²: 1, 2, 3, 6, x, 2x, 3x, 6x, x², 2x², 3x², and 6x².

For 5x: 1, 5, x, and 5x.

For 6: 1 and 6.

3. Combine the factors from each term to get the full list of factors:

The full list of factors is: 1, 2, 3, 5, 6, 10, 15, 30, x, 2x, 3x, 5x, 6x, 10x, 15x, 30x, x², 2x², 3x², 5x², and 6x².

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Answer:

Step-by-step explanation:B

Answer:

D is the answer

Step-by-step explanation:

The value of k where the roots of the quadratic equation are real and equal is k = 9.

Option B is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

3x + 6 = 9 is an equation.

We have,

3x² + 2kx + 27 = 0

This is in the form of ax² + bx + c.

a = 3

b = 2k

c = 27

Now,

The roots are real and equal.

D = 0

D = b² - 4ac

Now,

b² - 4ac = 0

(2k)² - 4 x 3 x 27 = 0

4k² - 324 = 0

4k² = 324

k² = 324/4

k² = 81

k = √81

k = ±9

Thus,

The value of k is 9 0r -9

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Answer:

Step 2

Step-by-step explanation:

3x-24=5x

he did -3x on each side but added 3x to 5x instead of subtracting

Justin and Tim have played the same game a total of 10 times.

There are 10 players in Crestwood Elementary's four-square league, including Justin and Tim. Over the course of a semester, each possible match-up of five players occurs once. This means that Justin and Tim have played the same game a total of 10 times.

To calculate this, we need to consider the different combinations of players that can make up a five-person game. There are a total of 10! (10 factorial) permutations of the 10 players, meaning that each possible game is made up of different players.

This includes the combination of Justin and Tim, which occurs 10 times. Therefore, Justin and Tim have played the same game 10 times during the semester.

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A root of degree 5 is called a fifth root.

A radicand, or nth root, in mathematics is a number r that, when raised to the power n, equals the number x: rn = x, where n is a positive integer, also known as the degree of the root. A root of degree 5 is called a fifth root.

The required number is the result of multiplying a fifth root by itself five times.

Due to the fact that two multiplied by itself five times is 32, 532 = 2

Seven multiplied by itself five times is 16807.

Hence 516807 Equals 7 since 7 x 7 x 7 x 7 = 16807.

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The Positive Infinity is called the Lemniscate.

Infinity:

Infinity is anything greater than infinity, infinity, or any natural number. Often represented by the infinity symbol {\displaystyle \infty }\infty. The most interesting point about infinity is – ∞ < x < ∞. This is a mathematical shorthand for negative infinity less than real and positive infinity greater than real. where 'x' represents a real number.

Positive Infinity:

Numbers approach negative infinity as they move left on the number line and positive infinity as they move right on the number line. Negative infinity is always less than any number and positive infinity is always greater than any number.

Positive Infinity: Differs from mathematical infinity in that:

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Answer: x=-3, y=2

Step-by-step explanation: to solve using substitution, we need either x or y by itself in one equation. let's use x from the 2nd equation since it doesn't have a constant.

so the first equation stays the same at 2x+3y=0

for the second equation, rearrange to solve for x by moving the 2y to the right -> x=2y-1

then, we can plug in this x value to x in the first equation to get,

2(2y-1)+3y=0

then solve for y,

4y-2+3y=0

4y-3y=2

y=2

then to solve for x, plug in y in either of the initial equations. lets choose the second one.

so you'd have,

x+2(2)=1

x+4=1

x=-3

A dataset's mean (also known as the arithmetic mean, which differs from the geometric mean) is calculated by dividing the sum of all values by the total number of values. The term "average" is frequently used to describe this measure of central tendency.

What does the mode mean?

By summing all the numbers in a data collection and dividing by the total number of values in the set, the mean (average) of the data set can be determined.

            A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently.

What are the mean, median, and example?

The most frequent number, or the one that happens the most frequently, is known as the mode.

                    Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.

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Answer:

----------------------

Given options:

A) f(x) = 2x² + 5x − 12,

B) f(x) = 2x + 5,

C) f(x) = 2(x + 5)² − 12,

D) f(x) = 2(x + 5)(x − 12),

The percent of decrease, in the number of tickets that he has, is 20%.

The term "percentage" refers to any ratio or number that may be expressed as a fraction of 100. And the sign "%" is used to denote it.

Given, Tyrese bought 35 tickets to use at a carnival.

He used some of the tickets to ride the bumper cars and play some games. Now he has 28 tickets left.

That means,

he used 35 - 28 = 7 tickets.

To find the percentage of decrease:

Let m be the percentage of decrease.

Then,

7 = m x 35 / 100

m = 700 / 35

m = 20

Therefore, the percentage of decrease is 20%

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Since the provided expression [tex]2x^{2} +8x-x^{2} +16[/tex]  is quadratic, the solution of x are "-4" and "-4" .

Any equation in algebra that can be written in standard form as where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation.

An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles. An expression in mathematics is made up of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) You may think of expressions as being comparable to phrases.

Expression:    [tex]2x^{2} +8x-x^{2} +16[/tex]  

This means that you need to add the two x² terms and the two x terms, which gives you [tex]x^{2} +8x+16[/tex].

Next, you can simplify the equation by taking out a common factor of x from the consecutive two terms, which gives you  [tex]x(x + 4) +4(x+4).[/tex]

Then the equation becomes [tex](x + 4)(x+4).[/tex] which on solving i.e [tex](x+4)=0[/tex]

you will get x= -4 and -4.

Therefore the solution to this equation is x = -4 or x = -4.

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By the distribution property 6(1/2 + 2/3+3/4) = 23/2    

According to the distributive property, multiplying the difference or aum of numbers will be equivalent to multiplying the individual parts of the difference or sum.

The expression that can explain the above rule is A ( B+ C) = AB + AC

Here we have

6(1/2 + 2/3+3/4)    

By distribution property

=> 6(1/2 + 2/3+3/4)  

= 6(1/2) + 6(2/3)+ 6(3/4)  

= 3 + 2(2) + 3(3/2)          

= 3 + 4 + 9/2

= 7 + 9/2

=  (14+9)/2        [ add 7 and 9/2 by taking 2 as LCM ]

=  23/2  

Therefore,

By the distribution property 6(1/2 + 2/3+3/4) = 23/2    

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Complete Question:

1. Find the value of this fraction computation: 6(1/2 + 2/3+3/4)    

Answer:

The greatest common monomial factor of [tex]3x^2 - 6x[/tex] is 3x.

Step-by-step explanation:

To understand what the problem is asking us, let's define the terms.

⭐What does greatest common factor mean?

⭐What does monomial mean?

Thus, we need to find the greatest term that is a factor of [tex]3x^2 -6x[/tex]. This term must be a monomial.

First, divide both terms of the polynomial [tex]3x^2 -6x[/tex] by the largest term that is common between the terms.

[tex]3x[/tex] is the largest term.

[tex]3x[/tex] is also a monomial.

Therefore, [tex]3x[/tex] is the greatest common monomial factor.

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The solution for z in the equation is z = 3y - 1/16(5y + 7x)⁴

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

2(5y+7x)²=8√(3y-z)

Rewrite the equation

So, we have the following representation

2(5y + 7x)² = 8√(3y - z)

Take the square of both sides

This gives

4(5y + 7x)⁴ = 64(3y - z)

Divide both sides by 64

1/16(5y + 7x)⁴ = 3y - z

So, we have

z = 3y - 1/16(5y + 7x)⁴

Hence, the solution is z = 3y - 1/16(5y + 7x)⁴

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Answer: 1.5 and 20

Step-by-step explanation:

The volume of the rectangular prism is 3 times the volume of the rectangular pyramid.

Consider a rectangular prism with length l, width w and height h.

The volume of the rectangular prism is:

volume of a rectangular prism = length * width * height

V1 = l * w * h

Consider a rectangular pyramid with rectangular base of length l1, width w1 and the height of pyramid is h1.

The volume of the rectangular pyramid is:

volume of a rectangular pyramid = 1/3 * base area * height

Here, base area = length * width

                          = l1 * w1

So, volume of a rectangular pyramid = 1/3 * base area * height

V2 = 1/3 * l1 * w1 * h1

We have been given a rectangular prism and a rectangular pyramid have equal length, width and height .

l1 = l, w1 = w and h1 = h

so, V2 = 1/3 * l * w * h

V2 = 1/3 * V1

V1 = 3 * V2

volume of a rectangular prism = 3 * volume of a rectangular pyramid

Therefore, the relationship between the volumes of a rectangular prism and a rectangular pyramid whose length width and height are all equal:

volume of rectangular prism = 3 * volume of a rectangular pyramid

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