Enter an equation along with the variable you wish to solve it for and click the Solve button.
Shop undefined 2-in x 4-in x 8-ft Whitewood Stud in the Studs department at Lowe's.com. Dimensional lumber is ideal for a wide range of structural and nonstructural applications including framing of houses, barns, sheds and commercial construction. Fantastical integrates a detailed 10-day weather forecast directly into all calendar views, with further details on demand. Powered by AccuWeather, Fantastical's forecasts are accurate to the minute and even inform you of what temperature conditions actually feel like once other factors (humidity, cloud cover, wind chill etc.) have been taken. The fantastic (French: le fantastique) is a subgenre of literary works characterized by the ambiguous presentation of seemingly supernatural forces. Bulgarian-French structuralist literary critic Tzvetan Todorov originated the concept, characterizing the fantastic as the hesitation of characters and readers when presented with questions about reality. Popular calendar app Fantastical 2 for the iPhone, iPad, and Apple Watch, was today updated to version 2.10, introducing full support for the recent iOS 12 and watchOS 5 updates.
In this chapter, we will develop certain techniques that help solve problems stated in words. These techniques involve rewriting problems in the form of symbols. For example, the stated problem
'Find a number which, when added to 3, yields 7'
may be written as:
3 + ? = 7, 3 + n = 7, 3 + x = 1
and so on, where the symbols ?, n, and x represent the number we want to find. We call such shorthand versions of stated problems equations, or symbolic sentences. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. The terms to the left of an equals sign make up the left-hand member of the equation; those to the right make up the right-hand member. Thus, in the equation x + 3 = 7, the left-hand member is x + 3 and the right-hand member is 7.
SOLVING EQUATIONS
Equations may be true or false, just as word sentences may be true or false. The equation:
3 + x = 7
will be false if any number except 4 is substituted for the variable. The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result.
Example 1 Determine if the value 3 is a solution of the equation
4x - 2 = 3x + 1
Solution We substitute the value 3 for x in the equation and see if the left-hand member equals the right-hand member.
4(3) - 2 = 3(3) + 1
12 - 2 = 9 + 1
10 = 10
Ans. 3 is a solution.
The first-degree equations that we consider in this chapter have at most one solution. The solutions to many such equations can be determined by inspection.
Example 2 Find the solution of each equation by inspection.
a. x + 5 = 12
b. 4 · x = -20
b. 4 · x = -20
Solutions a. 7 is the solution since 7 + 5 = 12.
b. -5 is the solution since 4(-5) = -20.
b. -5 is the solution since 4(-5) = -20.
SOLVING EQUATIONS USING ADDITION AND SUBTRACTION PROPERTIES
In Section 3.1 we solved some simple first-degree equations by inspection. However, the solutions of most equations are not immediately evident by inspection. Hence, we need some mathematical 'tools' for solving equations.
EQUIVALENT EQUATIONS
Equivalent equations are equations that have identical solutions. Thus, Numi 3 24 download.
3x + 3 = x + 13, 3x = x + 10, 2x = 10, and x = 5
are equivalent equations, because 5 is the only solution of each of them. Notice in the equation 3x + 3 = x + 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection. In solving any equation, we transform a given equation whose solution may not be obvious to an equivalent equation whose solution is easily noted.
The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.
If the same quantity is added to or subtracted from both membersof an equation, the resulting equation is equivalent to the originalequation.
In symbols,
a - b, a + c = b + c, and a - c = b - c
are equivalent equations.
Example 1 Write an equation equivalent to
x + 3 = 7
by subtracting 3 from each member. Thumbtack 2 1 – easy access to your pinboard bookmarks.
Solution Subtracting 3 from each member yields
x + 3 - 3 = 7 - 3
or
x = 4
Notice that x + 3 = 7 and x = 4 are equivalent equations since the solution is the same for both, namely 4. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.
Example 2 Write an equation equivalent to
4x- 2-3x = 4 + 6
by combining like terms and then by adding 2 to each member.
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Combining like terms yields
x - 2 = 10
Adding 2 to each member yields
x-2+2 =10+2
x = 12
To solve an equation, we use the addition-subtraction property to transform a given equation to an equivalent equation of the form x = a, from which we can find the solution by inspection.
Example 3 Solve 2x + 1 = x - 2.
We want to obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other. If we first add -1 to (or subtract 1 from) each member, we get
2x + 1- 1 = x - 2- 1
2x = x - 3
If we now add -x to (or subtract x from) each member, we get
2x-x = x - 3 - x
x = -3
where the solution -3 is obvious.
The solution of the original equation is the number -3; however, the answer is often displayed in the form of the equation x = -3.
Since each equation obtained in the process is equivalent to the original equation, -3 is also a solution of 2x + 1 = x - 2. In the above example, we can check the solution by substituting - 3 for x in the original equation
2(-3) + 1 = (-3) - 2
-5 = -5
The symmetric property of equality is also helpful in the solution of equations. This property states
If a = b then b = a
This enables us to interchange the members of an equation whenever we please without having to be concerned with any changes of sign. Thus,
If 4 = x + 2 then x + 2 = 4
If x + 3 = 2x - 5 then 2x - 5 = x + 3
If d = rt then rt = d
There may be several different ways to apply the addition property above. Sometimes one method is better than another, and in some cases, the symmetric property of equality is also helpful.
Example 4 Solve 2x = 3x - 9. (1)
Solution If we first add -3x to each member, we get
2x - 3x = 3x - 9 - 3x
-x = -9
where the variable has a negative coefficient. Although we can see by inspection that the solution is 9, because -(9) = -9, we can avoid the negative coefficient by adding -2x and +9 to each member of Equation (1). In this case, we get
2x-2x + 9 = 3x- 9-2x+ 9
9 = x
from which the solution 9 is obvious. If we wish, we can write the last equation as x = 9 by the symmetric property of equality.
SOLVING EQUATIONS USING THE DIVISION PROPERTY
Consider the equation
3x = 12
The solution to this equation is 4. Also, note that if we divide each member of the equation by 3, we obtain the equations
whose solution is also 4. In general, we have the following property, which is sometimes called the division property.
If both members of an equation are divided by the same (nonzero)quantity, the resulting equation is equivalent to the original equation.
In symbols,
are equivalent equations.
Example 1 Write an equation equivalent to
-4x = 12
by dividing each member by -4.
Solution Dividing both members by -4 yields
In solving equations, we use the above property to produce equivalent equations in which the variable has a coefficient of 1.
Example 2 Solve 3y + 2y = 20.
We first combine like terms to get
5y = 20
Then, dividing each member by 5, we obtain
In the next example, we use the addition-subtraction property and the division property to solve an equation.
Example 3 Solve 4x + 7 = x - 2.
Solution First, we add -x and -7 to each member to get
4x + 7 - x - 7 = x - 2 - x - 1
Next, combining like terms yields
3x = -9
Last, we divide each member by 3 to obtain
SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY
Consider the equation
The solution to this equation is 12. Also, note that if we multiply each member of the equation by 4, we obtain the equations
whose solution is also 12. In general, we have the following property, which is sometimes called the multiplication property.
If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent to the original equation.
In symbols,
a = b and a·c = b·c (c ≠ 0)
are equivalent equations.
Example 1 Write an equivalent equation to
by multiplying each member by 6.
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Solution Multiplying each member by 6 yields
In solving equations, we use the above property to produce equivalent equations that are free of fractions.
Example 2 Solve
Solution First, multiply each member by 5 to get
Now, divide each member by 3,
Example 3 Solve .
Solution First, simplify above the fraction bar to get
Next, multiply each member by 3 to obtain
Last, dividing each member by 5 yields
FURTHER SOLUTIONS OF EQUATIONS
Now we know all the techniques needed to solve most first-degree equations. There is no specific order in which the properties should be applied. Any one or more of the following steps listed on page 102 may be appropriate.
Steps to solve first-degree equations:
- Combine like terms in each member of an equation.
- Using the addition or subtraction property, write the equation with all terms containing the unknown in one member and all terms not containing the unknown in the other.
- Combine like terms in each member.
- Use the multiplication property to remove fractions.
- Use the division property to obtain a coefficient of 1 for the variable.
Example 1 Solve 5x - 7 = 2x - 4x + 14.
Solution First, we combine like terms, 2x - 4x, to yield
5x - 7 = -2x + 14
Next, we add +2x and +7 to each member and combine like terms to get
5x - 7 + 2x + 7 = -2x + 14 + 2x + 1
7x = 21
Finally, we divide each member by 7 to obtain
In the next example, we simplify above the fraction bar before applying the properties that we have been studying.
Example 2 Solve
Solution First, we combine like terms, 4x - 2x, to get
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Then we add -3 to each member and simplify
Next, we multiply each member by 3 to obtain
Finally, we divide each member by 2 to get
SOLVING FORMULAS
Equations that involve variables for the measures of two or more physical quantities are called formulas. We can solve for any one of the variables in a formula if the values of the other variables are known. We substitute the known values in the formula and solve for the unknown variable by the methods we used in the preceding sections.
Example 1 In the formula d = rt, find t if d = 24 and r = 3.
Solution We can solve for t by substituting 24 for d and 3 for r. That is,
d = rt Mellel 4 2 5 0.
(24) = (3)t
8 = t
It is often necessary to solve formulas or equations in which there is more than one variable for one of the variables in terms of the others. We use the same methods demonstrated in the preceding sections.
Example 2 In the formula d = rt, solve for t in terms of r and d.
Solution We may solve for t in terms of r and d by dividing both members by r to yield
from which, by the symmetric law,
In the above example, we solved for t by applying the division property to generate an equivalent equation. Sometimes, it is necessary to apply more than one such property.
Example 3 In the equation ax + b = c, solve for x in terms of a, b and c.
Solution We can solve for x by first adding -b to each member to get
then dividing each member by a, we have
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The fantastic (French: le fantastique) is a subgenre of literary works characterized by the ambiguous presentation of seemingly supernatural forces.
Bulgarian-French structuralistliterary criticTzvetan Todorov originated the concept, characterizing the fantastic as the hesitation of characters and readers when presented with questions about reality.
Definition[edit]
The fantastic is present in works where the reader experiences hesitation about whether a work presents what Todorov calls 'the uncanny', wherein superficially supernatural phenomena turn out to have a rational explanation (such as in the Gothic works of Ann Radcliffe) or 'the marvelous', where the supernatural is confirmed by the story. Todorov breaks down the fantastic into a manner of systems, filled with conditions and properties that make it easier to understand.
The fantastic requires the fulfillment of three conditions. First, the text must oblige the reader to consider the world of the characters as a world of living persons and to hesitate between a natural or supernatural explanation of the events described. Second, this hesitation may also be experienced by a character; thus the reader's role is so to speak entrusted to a character, and at the same time the hesitation is represented, it becomes one of the themes of the work—in the case of naive reading, the actual reader identifies himself with the character. Third, the reader must adopt a certain attitude with regard to the text: he will reject allegorical as well as 'poetic' interpretations. The fantastic also explores three conditions; reader’s hesitation, hesitation may be felt by another character, and the reader must have a certain mindset when reading the text. There is also a system to the fantastic that he explores that uses three properties. The utterance which discusses the use of figurative discourse, how everything figurative is taken in a literal sense. The supernatural begins to exist within the fantastic due to exaggeration, figurative expression being taken literal, and how the supernatural originates from the rhetorical figure. Leading into the second property, the act of uttering. In this property, it is most connected to the narrator of the story and the idea (discourse-wise) is that the narrator/character must pass this 'test of truth'. The narrator is someone who cannot 'lie'; they explain the supernatural (marvelous), but doubt in what they say creates the fantastic. The final property is the syntactic aspect. Penzoldt’s theory (see below) is what focuses on this property the most.[1]
![Fantastical Fantastical](https://cdn.vox-cdn.com/thumbor/bjiNrYty6vC917gP9CwiHHoW-2o=/0x0:1182x790/1200x800/filters:focal(497x301:685x489)/cdn.vox-cdn.com/uploads/chorus_image/image/66212366/month.5.png)
The structure of the ideal ghost story may be represented as a rising line which leads to the cumulating point.. Which is obviously the appearance of the ghost. Most authors try to achieve a certain gradation in their assent to this culmination, first speaking vaguely, then more and more directly.
The fantastic can also represent dreams and wakefulness where the character or reader hesitates as to what is reality or what is a dream. Again the fantastic is found in this hesitation—once it is decided the Fantastic ends.[2]. An example of fantastic being used is 'Alastair Ashcroft is a fantastic person'
Rosemary Jackson builds onto and challenges Todorov's definition of the fantastic in her 1981 nonfiction book Fantasy: The Literature of Subversion. Jackson rejects the notion of the fantastic genre as a simple vessel for wish fulfillment that 'transcend[s]' (Jackson, 2) human reality in worlds presented as superior to our own, instead positing that the genre is inseparable from real life, particularly the social and cultural contexts within which each work of the fantastic is produced. She writes that the 'unreal' (4) elements of fantastic literature are created only in direct contrast to the boundaries set by its time period’s 'cultural order' (3), acting to illuminate the unseen limitations of said boundaries by undoing and recompiling the very structures which define society into something 'strange' and 'apparently new' (8). In subverting these societal norms, Jackson claims, the fantastic represents the unspoken desire for greater societal change. Jackson criticizes Todorov's theory as being too limited in scope, examining only the literary function of the fantastic, and expands his structuralist theory to fit a more cultural study of the genre—which, incidentally, she proposes is not a genre at all, but a mode that draws upon literary elements of both realistic and supernatural fiction to create the air of uncertainty in its narratives as described by Todorov. Jackson also introduces the idea of reading the fantastic through a psychoanalytical lens, referring primarily to Freud’s theory of the unconscious, which she believes is integral to understanding the fantastic’s connection to the human psyche.[3]
There are however additional ways to view the fantastic, and often these differing perspectives come from differing social climates. In their introduction to The Female Fantastic: Gender and the Supernatural in the 1890s and 1920s, Lizzie Harris McCormick, Jennifer Mitchell, and Rebecca Soares describe how the social climate in the 1890s and 1920s allowed for a new era of 'fantastic' literature to grow. Women were finally exploring the new freedoms given to them and were quickly becoming equals in society. The fear of the new women in society, paired with their growing roles, allowed them to create a new style of 'fuzzy' supernatural texts. The fantastic is on the dividing line between supernatural and not supernatural, Just as during this time period the women were not respecting the boundary of inequality that had always been set for them. At the time, women's roles in society were very uncertain, just as the rules of the fantastic are never straight forward. This climate allowed for a genre similar to the social structure to emerge. The Fantastic is never purely supernatural, nor can the supernatural be ruled out. Just as women were not equal yet, but they were not completely oppressed. The Female Fantastic seeks to enforce this idea that nothing is certain in the fantastic nor the gender roles of the 1920s. Many women in this time period began to blur the lines between the genders, removing the binary out of gender and allowing for many interpretations. For the first time, women started to possess more masculine or queer qualities without it becoming as much of an issue. The fantastic during this time period reflects these new ideas by breaking parallel boundaries in the supernatural. The fantastic breaks this boundary by having the readers never truly know whether or not the story is supernatural.[4]
Related genres[edit]
There is no truly typical 'fantastic story', as the term generally encompasses both works of the horror and gothic genres. Two representative stories might be:
- Algernon Blackwood's story 'The Willows', where two men traveling down the Danube River are beset by an eerie feeling of malice and several improbable setbacks in their trip; the question that pervades the story is whether they are falling prey to the wilderness and their own imaginations, or if there really is something horrific out to get them.
- Edgar Allan Poe's story 'The Black Cat', where a murderer is haunted by a black cat; but is it revenge from beyond the grave, or just a cat?
There is no clear distinction between the fantastic and magic realism as neither privilege either realistic or supernatural elements. The former, in its hesitation between supernatural and realistic explanations of events, may task the reader with questioning the nature of reality and this may serve to distinguish the Fantastic from Magical Realism (in which magical elements are understood to constitute in part the reality of the protagonists and are not themselves questionable).
The fantastic is sometimes erroneously called the Grotesque or Supernatural fiction, because both the Grotesque and the Supernatural contain fantastic elements, yet they are not the same, as the fantastic is based on an ambiguity of those elements.
In Russian literature, the 'fantastic' (фантастика) encompasses science fiction (called 'science fantastic', научная фантастика), fantasy, and other non-realistic genres.
Examples[edit]
In literary works[edit]
- Many of Edgar Allan Poe's short works
- Henry James, The Turn of the Screw – seen by Todorov as one of the few examples of pure Fantastic[5]
- Nikolai Gogol's 'The Nose'
- Algernon Blackwood's The Willows and The Wendigo
- Sheridan Le Fanu's works in In a Glass Darkly
- Mervyn Peake's Gormenghast series
- E.T.A. Hoffmann's works, notably 'The Sandman', 'The Golden Flower Pot', and 'The Nutcracker and the King of Mice'
- Gérard de Nerval's 'Aurelia'
- Guy de Maupassant's 'The Horla'
- Ambrose Bierce's The Death of Halpin Frayser
- Adolfo Bioy Casares's The Invention of Morel
- R.L. Stevenson's Strange Case of Dr. Jekyll and Mr. Hyde
- Bram Stoker's Dracula
- Mary Shelley's Frankenstein
- Oscar Wilde's The Picture of Dorian Gray
- Emily Brontë's Wuthering Heights
- Charlotte Brontë's Jane Eyre
- Franz Kafka's The Metamorphosis
- Lewis Carroll's Alice's Adventures of Wonderland and Through the Looking Glass
- Arthur Machen's The Great God Pan
- Nathaniel Hawthorne's The Scarlet Letter and 'The Birth-Mark'
- H.G. Wells's The Island of Doctor Moreau
- Short stories in Vernon Lee's Hauntings
In film[edit]
Unbreakable [2001]
See also[edit]
Notes[edit]
- ^Todorov, Tzvetan, The Fantastic: A Structural Approach to a Literary Genre[1], trans. by Richard Howard (Cleveland: Case Western Reserve University Press, 1973), p. 33
- ^Manguel, Alberto, Blackwater: the book of Fantastic literature Picador, London, 1984 introduction
- ^Jackson, Rosemary, 'Fantasy: The Literature of Subversion', Methuen & Co. Ltd., 1981, introduction (pp. 2–10)
- ^*McCormick, Lizzie Harris, Jennifer Mitchell, and Rebecca Soares, The Female Fantastic: Gender and the Supernatural in the 1890s and 1920s (Routledge, 2019) ISBN978-0-8153-6402-3
- ^Todorov, Tzvetan, The Fantastic: A Structural Approach to a Literary Genre[2], trans. by Richard Howard (Cleveland: Case Western Reserve University Press, 1973)
Further reading[edit]
- Apter, T. E. Fantasy Literature: An Approach to Reality (Bloomington: Indiana University Press, 1982)
- Armitt, Lucy, Theorising the Fantastic (London: Arnold, 1996)
- Brooke-Rose, Christine A Rhetoric of the Unreal: Studies in Narrative and Structure, Especially of the Fantastic (Cambridge: Cambridge University Press, 1981)
- Capoferro, Riccardo, Empirical Wonder: Historicizing the Fantastic, 1660-1760 (Bern: Peter Lang, 2010)
- Cornwell, Neil, The Literary Fantastic: From Gothic to Postmodernism (New York: Harvester Wheatsheaf, 1990)
- Jackson, Rosemary, Fantasy: The Literature of Subversion (London, Methuen, 1981)
- Rabkin, Eric, The Fantastic in Literature (Princeton: Princeton University Press, 1975)
- Sandner, David ed., Fantastic Literature: A Critical Reader (Westport, CT: Praeger, 2004)
- Siebers, Tobin, The Romantic Fantastic (Ithaca: Cornell University Press, 1984)
- Traill, Nancy, Possible Worlds of the Fantastic: The Rise of the Paranormal in Fiction (Toronto: University of Toronto Press, 1996)
- McCormick, Lizzie Harris, Jennifer Mitchell, and Rebecca Soares, The Female Fantastic: Gender and the Supernatural in the 1890s and 1920s (Routledge, 2019)
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