Find the value of x
x + x + (2x + 5) + (2x + 7)

Answer:

Step-by-step explanation:

Let x be the number of children Grace babysits. Then, she will earn $50 + $5 per child, or a total of $50 + 5x. To make at least $63, we can write an inequality as follows:

50 + 5x >= 63

To solve for x, we can subtract 50 from both sides:

5x >= 13

And finally, divide both sides by 5:

x >= 2.6

Since Grace needs to babysit at least 2 whole children, she needs to babysit at least 3 children to make $63 or more.

Answer:

Step-by-step explanation:

Well if you do this [tex]$50 + $5=x\\50+5= 55+50+5=110[/tex]you will only have to babysit 2 kids and you will have leftover money.

[tex]65-110=x 65-110=153[/tex]

Answer: a/c

Step-by-step explanation: cancel out the common factor b. to find the excluded values would be b = 0, c = 0

The values are 0,27,27,1000 and dependent values.

A function is a mathematical rule that assigns a unique output to each input value. The input values are called the independent variables, and the output values are called the dependent variables. The relationship between the input and output values is defined by a mathematical expression or a set of instructions.

f(0) = |0|^3 = 0^3 = 0

f(-3) = |-3|^3 = 3^3 = 27

f(3) = |3|^3 = 3^3 = 27

f(-10) = |-10|^3 = 10^3 = 1000

f(a-3) = |a-3|^3 = (a-3)^3 if a >= 3, and -(a-3)^3 if a < 3

f(ah) = |ah|^3 = (ah)^3 if h >= 0, and -(ah)^3 if h < 0

So, depending on the values of a and h, the value of f(a-3) and f(ah) will change. For example, if a = 5 and h = -1, then f(a-3) = f(2) = |2|^3 = 8 and f(ah) = f(-5) = |-5|^3 = 125.

Therefore, The values are 0,27,27,1000 and dependent values.

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The values are 0,27,27,1000 and dependent values.

A function is a mathematical rule that assigns a unique output to each input value. The input values are called the independent variables, and the output values are called the dependent variables. The relationship between the input and output values is defined by a mathematical expression or a set of instructions.

f(0) = |0|³= 0³ = 0

f(-3) = |-3|³ = 3³ = 27

f(3) = |3|³ = 3³ = 27

f(-10) = |-10|³ = 10³ = 1000

f(a-3) = |a-3|³ = (a-3)³ if a >= 3, and -(a-3)³ if a < 3

f(ah) = |ah|³ = (ah)³ if h >= 0, and -(ah)³ if h < 0

So, depending on the values of a and h, the value of f(a-3) and f(ah) will change. For example, if a = 5 and h = -1, then f(a-3) = f(2) = |2|³ = 8 and f(ah) = f(-5) = |-5|³ = 125.

Therefore, The values are 0,27,27,1000 and dependent values.

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The center is called rectangle point X. The distance from the corner of the rectangle to the center (BX) can be found using the Pythagorean theorem is 21.74 to the nearest tenth be 21.7.

The Pythagorean theorem defined that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the right triangle is constructed by sides AB, BC, and BX, with BX being the hypotenuse. So, we can apply the Pythagorean theorem as follows:

        BX² = AB² + BC²

Plugging in the values for AB and BC:

        BX²= 12² + 18²

        BX² = 144 + 324

        BX² = 468

Finally, taking the square root of both sides gives us the length of BX:

        BX = √(468)

        BX = 21.74 to the nearest tenth = 21.7

The description above involves finding the length of the diagonal of a rectangle that goes from one corner of the rectangle to its center. This length is similar to as BX in the definition. To find BX, the Pythagorean theorem is used.

The Pythagorean theorem defined that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the right triangle is constructed by sides AB, BC, and BX, with BX being the hypotenuse.

By substituting the values of AB and BC, the equation BX² = AB² + BC² can be solved to find the length of BX. Finally, taking the square root of the solution gives the length of BX to the nearest tenth.

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The solution is Option A.

The value of the equation is x = -13/15 and x = 0

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the quadratic equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

15x² + 13x = 0   be equation (1)

On simplifying the equation , we get

Subtracting 13x on both sides of the equation , we get

15x² = -13x

Divide by x on both sides of the equation , we get

15x = -13

Divide by 15 on both sides of the equation , we get

x = - ( 13/15 )

So , when x = 0

0 = 0

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

x = -13 ± √ ( 169 - 4 ( 15 ) ( 0 ) ) / 2 ( 15 )

x = -13 ± √ ( 169 - 0 ) / 30

x = ( -13 ± 13 ) / 30

So , we have 2 solutions to x ,

x = ( -13 - 13 ) / 30

x = -26 / 30

x = -13/15

And , x = ( -13 + 13 ) / 30

x = 0/30

x = 0

Hence , the solutions to the equation is x = -( 13/15 ) and x = 0

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Solving provided question, we can say that value of standard deviation will be 3.28634 and Mean (Average) = 5 and Variance (SD) = 10.8

Standard deviation is a statistic that expresses the variability or variance of a group of numbers. While a high standard deviation suggests that the values are more dispersed, a low standard deviation suggests that the values tend to be closer to the established mean. A measure of how dispersed the data are from the mean is the standard deviation (or ). When the standard deviation is low, the data tend to be grouped around the mean, and when it is large, the data are more dispersed. The average variability of the data set is measured as standard deviation. It reveals the average deviation of each score from the mean.

Total Numbers = 6

Mean (Average) = 5

SD = 3.28634

Variance (SD) = 10.8

Population SD = 3

Variance(Population SD) = 9

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The amount of the old monthly payment = $879

The amount of the new monthly payment = $724.5

Then the small amount of principal that has been paid = -17.57%

Given that,

Natan Nave bought a home with a 13% adjustable rate mortgage for 20 years.

He paid $11. 72 monthly per thousand on his original loan.

At the end of 4 years he owes the bank $75,000.

Now that interest rates have gone down to 10%

The bank will renew the mortgage at this rate or Natan can pay $75,000.

Natan decides to renew and will pay $9. 66 monthly per thousand on this loan.

Per thousand value = 75000/1000 = 75

What was the old monthly payment,

$[tex]11.72*75[/tex] = $879

What is the new monthly payment,

$[tex]9.66*75[/tex] = $724.5

What is the percent decrease in his monthly payment (to the nearest tenth),

= [tex](\frac{9.66}{11.72} )-1 *100[/tex]

= -17.57%

Therefore,

The amount of the old monthly payment = $879

The amount of the new monthly payment = $724.5

Then the small amount of principal that has been paid = -17.57%

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A bottle of tomato juice contains 247. 5 mg of vitamin C and A bottle of grape juice contains 258 mg are in one serving of each type of juice is 45mg and 86mg.

A bottle of tomato juice contains 247. 5 mg of vitamin C and has 5. 5 servings so,

247.5/5.5 = 45 mg

A bottle of grape juice contains 258 mg of vitamin C and has 3 servings so,

258/3 = 86 mg

Water-soluble vitamin C, sometimes referred to as ascorbic acid and ascorbate, is present in citrus and other fruits and vegetables. It is also available as a dietary supplement and as an ingredient in topical "serums" used to cure melasma (dark pigment spots) and facial wrinkles. Scurvy is treated and prevented with it.

Vitamin C is a necessary nutrient that aids in collagen development, tissue healing, and the enzymatic synthesis of certain neurotransmitters. It is necessary for the operation of various enzymes and is crucial for the immune system. Additionally, it serves as an antioxidant. The majority of animals can produce their own vitamin C.

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Let's call the amount of Deep Thought granola mixed "x" (in pounds). The amount of Oat Dream granola mixed would then be 20 - x (in pounds).

We know that Deep Thought granola is 25% nuts and dried fruit and Oat Dream granola is 10% nuts and dried fruit. We also know that the desired mix of 20 pounds of granola should be 19% nuts and dried fruit.

Therefore, we can set up the following equation to find the amount of Deep Thought and Oat Dream granola to be mixed:

0.25x + 0.10(20 - x) = 0.19 * 20

Expanding the right side of the equation and solving for x, we get:

0.25x + 2 - 0.1x = 3.8

0.15x = 1.8

x = 12

So, we would mix 12 pounds of Deep Thought granola and 8 pounds of Oat Dream granola to get 20 pounds of granola that is 19% nuts and dried fruit.

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The way that  the Great Awakening lead to a new American government is that; The Great Awakening helped the colonists realize they needed their own national religion.

The Great Awakening was a period in history characterized by religious revival in the US. There were five major Great Awakenings, that occurred between the beginning of 18th and the end of the 20th century. However, the last of the Great Awakenings lasted until the year 2000.

During the years of 1730s and 1740s, it was clear that the English colonies in America went through the Great Awakening which was a religious resurgence. This movement came into play at a period when there was a very huge emphasis on the idea of secular reason and as a result religious zeal had lost its freshness.

The gospel was usually preached by Christian leaders from town to town to stir up interest in Christianity while emphasizing deliverance from sin. The outcome or result was a reinvigorated commitment to religion.

The Great Awakening, in the opinion of many historians, had a profound effect on the American culture as a whole as well as on different Christian denominations.

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Because the general solution involves an arbitrary constant of integration that can only be discovered by providing an extra beginning condition, the solution to this equation with the starting condition y(0) = 0 is not unique.

An algebraic equation, also known as a polynomial equation, is a mathematical equation of the form P=0, where P is a polynomial with coefficients in some field, most commonly the field of rational numbers. An algebraic equation is a mathematical statement that sets two expressions equal to each other. An algebraic equation is typically made up of a variable, coefficients, and constants.

Here,

The equation y' = xy is a separable differential equation, which means it can be solved for y in terms of x by separating and integrating the variables. The solution to this equation with the starting condition y(0) = 0 is not unique since the general solution contains an arbitrary constant of integration that can only be found by giving an additional initial condition.

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the angle bisector theorem formula yields the value x=10.

The angle bisector theorem states that RU/RT=US/ST or a/b = x/y. A line or ray known as an angle bisector divides an angle in a triangle into two identically sized parts.

The geometry concept known as the angle bisector theorem examines the ratios of the two segments that a line that bisects the opposing angle splits a triangle's side into. It contrasts their relative lengths with the ratios of the other two sides of the triangle.

Side ratio: RU/RT=US/ST

3x/40 = x+2/16

6x=5x +10

x=10

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The following equations are organized for P as a function of Q

1) P = 20-8Q

2) p = [tex]\dfrac{12-Q}{3}[/tex]

3) p = 9 - 4Q

A function in mathematics is, technically speaking, a relation between a set of inputs and a set of potential outputs, where each input is related to exactly one output. Examples of functions in mathematics typically range from integers to integers or from real numbers to real numbers.

Additionally, it is a relation or a process that links each element x of a set X to the codomain of the function and to a single element y of a different set Y (typically the same set).

1) Q = 20-P

Q + P = 20

P = 20-Q

2) Q=12-3P

Q +3P = 12

3P = 12-Q

p = [tex]\dfrac{12-Q}{3}[/tex]

P = 4 - 1/3Q

3) 8Q = 18 -2p

8Q + 2p = 18

2p = 18 - 8Q

p =  [tex]\dfrac{18-8Q}{2}[/tex]

p = 9 - 4Q

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Complete question:

Let length of vector is 5 then the vector in the direction of  ⟨−3,−1,5⟩ will be ⟨−15/√35,−5/√35,25/√35⟩.

What is a Vector?

Vectors, in Maths, are objects which have both, magnitude and direction. Magnitude defines the size of the vector. It is represented by a line with an arrow, where the length of the line is the magnitude of the vector and the arrow shows the direction. It is also known as Euclidean vector or Geometric vector or Spatial vector or simply “vector“.

Two vectors are said to equal if their magnitude and direction are the same. It plays an important role in Mathematics, Physics as well as in Engineering. According to vector algebra, a vector can be added to another vector, head to tail. The order of addition of two vectors does not matter, because the result will be the same.

Now,

The idea is based on a concept of scaling and similarity.

Any vector that "has the same direction" as

⟨−3,−1,5⟩

has all the coordinates proportional to this given vector and, therefore, can be described by coordinates

⟨−3f,−1f,5f⟩

where f is a scaling factor.

All we need now is to find a scaling factor that leads to a vector with the length 5

The length of a vector with coordinates

⟨−3f,−1f,5f⟩

equals to√9f^2+f^2+25f^2=√35f

So, if we want the length to be equal to 5, we should choose

f=5/√35

Hence,

          The coordinates of a vector with the same direction as  ⟨−3,−1,5⟩ but with the length 5 will be ⟨−15/√35,−5/√35,25/√35⟩.

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The value of angles ∠A and ∠B is equal to 40°.

Given is that ∠A = 5x − 5 and ∠B = 3x + 13

We can write -

∠A = ∠B            {Vertically opposite angles}

5x − 5 = 3x + 13

5x - 3x = 18

2x = 18

x = 9

So, we can write -

∠A = 5x − 5 = 40°

∠B = 3x + 13 = 40°

Therefore, the value of angles ∠A and ∠B is equal to 40°.

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The standard parameterization for the tangent line :

x = 12t + 18 , y = 3t + 13 , z = 54t + 54

parametric equation, a type of equation that employs an independent variable called a parameter (often denoted by t) and in which dependent variables are defined as continuous functions of the parameter and are not dependent on another existing variable.

as given,

r(t) = (2t^2) i + (3t + 4) j + (2t^3) k, t = t₀ = 3

so, [tex]x = 2t^{2}, y = 3t + 4 , z = 2t^{3}[/tex]

[tex]\frac{dx}{dt} = 4t, y=3, z=6t^{2}[/tex]

r'(t) = <4t, 3, [tex]6t^{2}[/tex]>

and at t =3,

r'(t=3) = <12, 3, 54>

we have to find the tangent line to the curve at t=3,

x = 18, y = 13, z = 54

for t=3, (x, y, z) = (18, 13, 54)

finally,

x = 12t + 18

y = 3t + 13

z = 54t + 54

Thus, the standard parameterization for the tangent line :

x = 12t + 18 , y = 3t + 13 , z = 54t + 54

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The standard parameterization for the line tangent to the curve is

x=12t+18 , y=3t+9 , z=54t+54  

In parameterization the variables are independent variables , whereas continuous functions are dependent variables , that is it is dependent on another functions. The given curve is r(t)=(2t^2)i+(3t+4)j+(2t^3)k and given t=t0=3

Given , r(t)=(2t^2)i+(3t+4)j+(2t^3)k

r'(t)=(4t)i+3j+(6t^2)k

and also given t=t0=3

r(t=t0=3) => (18 , 9 , 54)

r'(t=t0=3) => (12 , 3 , 54)

The standard parameterization for the tangent line can be given by using r(t) and r'(t) . so by that we came to know the required standard parameterization.

x=12t+18

y=3t+9

z=54t+54

Therefore, the about mentioned x, y, z are the standard parameterization for the tangent line of given curve.

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Answer: 32

Step-by-step explanation:

Answer:

She had 32 pairs.

Step-by-step explanation:

Subtract 2 from 18 and then times that answer by 2.

Leon found the sum of 5/6 and 2/3 to be 11/12, you can tell that his answer is incorrect without calculating by seeing that after doing denominator same we do not get 11 as addition.

The sum of 5/6 and 2/3 will be calculated as,

= 5/6 + 2/3

= 5/6 + 2(2)/3(2)

= 5/6 + 4/6

= 9/6

= 3/2

So, the sum of 5/6 and 2/3 is 9/6.

Therefore, you can tell that his answer is incorrect without calculating by seeing that the given answer is not equivalent to the 11/12.

Since a fraction has a numerator and a denominator that are separated by a bar, adding fractions is a little different than adding integers normally. If the denominators are equal, adding fractions can be done with ease. Like fractions are transformed to like fractions because they share a common denominator, which makes addition simpler.

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All such strings if no repeats are allowed, meaning each of A, B, or C is used once then the total number of combinations is 13.

An array is a collection of elements that are ordered and can be accessed through indices.

In this case, we are dealing with arrays of character strings, where each string is a sequence of letters. In this problem, there are three possibilities for the characters that can appear in each string: A, B, or C.

(a) Three character strings where repeats are allowed:

Tree:

List: AAA, AAB, ACC, ABA, ABC, ACA

Number of items on list: 6

(b) Three character strings where repeats are not allowed in a row:

List: AB, AC, BAC, BCA, CAB, CBA

Number of items on list: 6

(c) Three character strings with no repeats allowed:

Tree:

ABC

List: ABC

Number of items on list: 1

So, the total number of items on the list is calculated as,

=> 6 + 6 + 1.

=> 13

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The y-intercept represents the initial temperature, and the rate of change gives how the temperature changes per minute.

So the correct options are A and D.

Here we have a relation that shows how the temperature changes as a function of time.

So if F(x) is the temperature and x is the time, the y-intercept is given by:

F(0)

So the y-intercept is the initial temperature, the correct option is A.

And the rate of change tells us how the temperature changes as the time passes, so we can define the rate as "the change in temperature per minute" like on option D.

These are the correct options.

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Therefore , the solution of the given problem of rational numbers  comes out to be A) 0.75 can be written as 75/100.

Rational numbers are those that may be written as a ratios (or fraction) of 2n. A fraction with a nonzero denominator is said to be rational. A few instances of rational numbers are 1/2, 1/5, and 3/4. As a real function, "0" can also be stated in a number of different ways, such as 0/1, 0/2, etc 0/3. But 1/0, 2/0, 3/0, and so forth. The number seven makes sense. A rational number is produced when three integers are split. To get the rational number 7, divide whole 7 ps by the total number 1. Since 7 can be obtained by dividing two integers, it is referred to as a real function.

Here,

Given :

=> Any number that can be expressed as a straightforward fraction is considered rational. This is the shape of a fraction: => a/b

where the denominator is "b" and the numerator is "a". Since both are integers, b 0.

=>Simple fractions cannot be used to represent irrational values.

So,

A) 75/100 is equivalent to 0.75 rational.

B) √√3 is irrational.

C) √√√9 is irrational.

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The system of equation y = −8x − 9 and y = 6x + 5. So the point of the specified is (-1, -1).

These are the specified parameters:

y = −8x − 9 and y = 6x + 5

Substitute the equation

y = −8x − 9

6x + 5 = −8x − 9

6x + 8x = -9 - 5

14x = -14

  x = -14/14

  x = -1

Substitute x = -1 into the equation

y = 6x + 5

y = 6 (-1) + 5

y = -6 + 5

y = -1

So the poin (x, y) is (-1, -1).

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In the function f(x) = 3x = 2, when input is 3 output is 11, and when input is

10 output is 32.

A function can be defined as the outputs for a given set of inputs.

The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.

Suppose we have a function f(x) = 3x = 2.

Here, 'x' is the input, and f(x) is the output.

Now, When the input is 3,

f(3) = 3×3 + 2.

f(3) = 9 + 2.

f(3) = 11.

Again when the input is 10,

f(10) = 3×10 + 2.

f(10) = 30 + 2.

f(10) = 32.

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The value of the function (g o f) (x) = 12x + 49.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is typically represented as y = f. (x).

Given that the value of f(x) = x + 4 and g(x) = 12x + 1.

The value of (g o f)(x) is:

g(x) = 12x + 1

Substitute the value of f(x) in x for the function of g(x):

(g o f) (x) = 12(x + 4) + 1

(g o f) (x) = 12x + 48 +1

(g o f) (x) = 12x + 49

Hence, the value of the function (g o f) (x) = 12x + 49.

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The value of the function (gof) (x) would be 12x + 49.

The set of all x such that x is in the domain of g and g(x)) is in the domain of f is the domain of f composed with g, as you should be aware of. A composite function's domain is the set of inputs x in the domain of g for which g(x) is in the domain of the function.

A composite function is also known as f of g of x. It is formally represented as f(g(x)) or (f g)(x), which indicates that x = g(x) should be substituted in f. (x). It can alternatively be written as "f circle g of x." It is a process that merges two functions into one new function.

Let the given value be f(x) = x + 4 and g(x) = 12x + 1.

The value of (g o f)(x) is:

Substitute the value of f(x) in x for the function of g(x):

(g o f) (x) = 12(x + 4) + 1

simplifying the above equation, we get

(g o f) (x) = 12x + 48 +1

(g o f) (x) = 12x + 49

Therefore, the value of the function (gof) (x) be 12x + 49.

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Answer: There are 1,000,000 nanoseconds in a millisecond.

Answer:

The answer is 1000000.

Using Laplace Transformation, the initial value problem y" - 2y' + y = 0 , y(0) = 1 , y'(0) = 2 is y(t) = e^t + te^t.

The differential equation is y" - 2y' + y = 0 and y(0) = 1 , y'(0) = 2

As we know that the Laplace Equation is:

[tex]L(y^n(t)) = s^nf(s) - s^{n-1}y(0) - s^{n-2}y'(0)-\dots-sy^{n-2}(0)-y^{n-1}(0)[/tex]

Now applying the Laplace Transformation in the differential equation

L(y") - 2L(y') + L(y) = 0........(1)

Now finding the value of L(y") using the equation

L(y") = s^2f(s) - s(1) - 2

L(y") = s^2f(s) - s - 2

Now finding the value of L(y') using the equation

L(y') = sf(s) - 1

Now finding the value of L(y) using the equation

L(y) = f(s)

Now putting the values in the equation 1

s^2f(s) - s - 2 - 2(sf(s) - 1) + f(s) = 0

Now solving

s^2f(s) - s - 2 - 2sf(s) + 2 + f(s) = 0

s^2f(s) - s - 2sf(s) + f(s) = 0

Taking common f(s) in term terms of f(s)

(s^2 - 2s + 1)f(s) - s = 0

Add s on both side, we get

(s^2 - 2s + 1)f(s) = s

We can write (s^2 - 2s + 1) as (s - 1)^2

(s - 1)^2f(s) = s

Divide by (s - 1)^2 on both side, we get

f(s) = s/(s - 1)^2

The inverse Laplace of f(s) = s/(s - 1)^2 is

y(t) = e^t + te^t

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The equation of the axis of symmetry for the function will be x  = -3.

ax²+bx+c=0, with a not, equals to 0 is a quadratic equation, which is a second-order polynomial equation in a single variable. It has at least one solution because it is a second-order polynomial equation, which is guaranteed by the algebraic fundamental theorem. The answer could be simple or complicated.

Given,  the function f(x)= -x²- 6x - 8 Since the given equation is a parabola.

Rewrite the given equation in the vertex form.

f(x) = -x²- 6x - 8

f(x) = -(x² + 6x +8)

f(x) = -(x² + 2 * 3 *x + 9 -1)

f(x) = -(x +3)² +1

Compare this with the general equation of the vertex form

y = a(x - h)²  + k

We get,

a = -1

h = -3

k = 1

Thus the vertex of the given parabola will be (-3, 1)

Since the general form of the focus of the parabola (h, k +p)

Where,

k = 1/4a

k = -1/4

Thus, the coordinates of the focus of the parabola will be (-3, 3/4)

Since the equation of symmetry will be the equation of the line that passes through the points focus and vertex.

Thus, the equation will be x = -3

Therefore, the equation of the axis of symmetry for the function will be x  = -3.

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The value of the limit is [tex]\lim_{x \to \frac{\pi}{4} } \frac{tan(x) - 1}{x - \frac{\pi}{4} } = 2[/tex].

A general technique for assessing indeterminate forms like 0/0 or / is the L'Hospital rule. L'Hospital's rule is applied to calculate the limits of indeterminate forms for calculus derivatives. It is possible to use the L Hospital rule many times. This rule is still valid after being applied, and it will take any indefinite form. L'Hospital's Rule cannot be applied if the issue is not of the indeterminate types.

Given that the limit is:

[tex]\lim_{x \to \frac{\pi}{4} } \frac{tan(x) - 1}{x - \frac{\pi}{4} }[/tex]

When the limit is substituted the limit is in an indeterminate form.

Using the L'Hospital rule we have:

[tex]\lim_{x \to \frac{\pi}{4} } \frac{tan(x) - 1}{x - \frac{\pi}{4} }\\\\\lim_{x \to \frac{\pi}{4} } \frac{1 + tan^2(x)}{1 }\\[/tex]

Substitute the value of limit in the equation:

[tex]\lim_{x \to \frac{\pi}{4} } \frac{tan(x) - 1}{x - \frac{\pi}{4} } = 2[/tex]

Hence, the value of the limit is [tex]\lim_{x \to \frac{\pi}{4} } \frac{tan(x) - 1}{x - \frac{\pi}{4} } = 2[/tex].

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Answer:

x = -1

y = -7

Step-by-step explanation:

y=9x+2`

-2x+7y=-47  ==> isolate y

Equation 1: 7y = 2x - 47  ==> add 2x on both sides

Equation 2: y = 9x + 2

7 * (y = 9x + 2)

Equation 3: 7y = 63x + 14

Equation 1: 7y = 2x - 47

Subtract equation 1 from equation 3:

  7y = 63x + 14

- (7y = 2x - 47)

7y-7y = 63x-2x + 14-(-47)

0 = 61x + 14 + 47

0 = 61x + 61  ==> solve for x

-61 = 61x  ==> subtract 61 on both sides to isolate x

x = -1

Now solve for y

y = 9(-1) + 2  ==> plugin -1 for x in y=9x+2`

y = -9 + 2  ==> simplify

y = -7

`

After calculations, it is determined that the segments do not form a right-angled triangle.

we need to calculate the distances between the points as the side lengths of the triangle.

these side lengths must satisfy the Pythagoras principle :

c² = a² + b²

with c being the Hypotenuse (side opposite of the 90° angle) and the longest of the 3 sides, a and b being the legs.

the distance between 2 points is again calculated via Pythagoras, as the coordinate differences (= the legs) with the distance as Hypotenuse create right-angled triangles.

this is then caked the distance formula, but it is all Pythagoras again.

AB² = (xB - xA)² + (yB - yA)² = (6 - -2)² + (2 - 2)² =

= 8² + 0² = 8²

AB = 8

AC² = (0 - -2)² + (6 - 2)² = 2² + 4² = 4 + 16 = 20

AC = sqrt(20)

BC² = (0 - 6)² + (6 - 2)² = (-6)² + 4² = 36 + 16 = 52

BC = sqrt(52)

as sqrt(20) is a little bit more than 4, and sqrt(52) is a little bit more than 7, 8 is the longest side and Hypotenuse.

if this is a right-angled triangle, then

8² = sqrt(20)² + sqrt(52)²

must be true.

but

64 = 20 + 52 = 72 is wrong.

so, the segments do not form a right-angled triangle.

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