Introduction and Section 1, Bourgeois and Proletarians (Part 1)

Introduction and Section 1, Bourgeois and Proletarians (Part 1)

The Manifesto begins by announcing, "A spectre is haunting Europe--the spectre of Communism." All of the European powers have allied themselves against Communism, frequently demonizing its ideas. Therefore, the Communists have assembled in London and written this Manifesto in order to make public their views, aims and tendencies, and to dispel the maliciously implanted misconceptions. The Manifesto begins by addressing the issue of class antagonism. Marx writes, "The history of all hitherto existing society is the history of class struggles." Throughout history we see the oppressor and oppressed in constant opposition to each other. This fight is sometimes hidden and sometimes open. However, each time the fight ends in either a revolutionary reconstruction of society or in the classes' common ruin.

In earlier ages, we saw society arranged into complicated class structures. For example, in medieval times there were feudal lords, vassals, guild-masters, journeymen, apprentices and serfs. Modern bourgeois society sprouted from the ruins of feudal society. This society has class antagonisms as well, but it is also unique: class antagonisms have become simplified, as society increasingly splits into two rival camps--Bourgeoisie and Proletariat.

The Manifesto then shows how the modern bourgeoisie is the product of several revolutions in the mode of production and of exchange. The development of the bourgeoisie began in the earliest towns, and gained momentum with the Age of Exploration. Feudal guilds couldn't provide for increasing markets, and the manufacturing middle class took its place. However, markets kept growing and demand kept increasing, and manufacture couldn't keep up. This led to the Industrial Revolution. Manufacture was replaced by "Modern Industry," and the industrial middle class was replaced by "industrial millionaires," the modern bourgeois. With these developments, the bourgeoisie have become powerful, and have pushed medieval classes into the background. The development of the bourgeoisie as a class was accompanied by a series of political developments. With the development of Modern Industry and the world-market, the bourgeoisie has gained exclusive political sway. The State serves solely the bourgeoisie's interests.

Historically, the bourgeoisie has played a quite revolutionary role. Whenever it has gained power, it has put to an end all "feudal, patriarchal, idyllic relations." It has eliminated the relationships that bound people to their superiors, and now all remaining relations between men are characterized by self-interest alone. Religious fervor, chivalry and sentimentalism have all been sacrificed. Personal worth is now measured by exchange value, and the only freedom is that of Free Trade. Thus, exploitation that used to be veiled by religious and political "illusions" is now direct, brutal and blatant. The bourgeoisie has changed all occupations into wage-laboring professions, even those that were previously honored, such as that of the doctor. Similarly, family relations have lost their veil of sentimentality and have been reduced to pure money relations.

In the past, industrial classes required the conservation of old modes of production in order to survive. The bourgeoisie are unique in that they cannot continue to exist without revolutionizing the instruments of production. This implies revolutionizing the relations of production, and with it, all of the relations in society. Thus, the unique uncertainties and disturbances of the modern age have forced Man to face his real condition in life, and his true relations with others.

Because the bourgeoisie needs a constantly expanding market, it settles and establishes connections all over the globe. Production and consumption have taken on a cosmopolitan character in every country. This is true both for materials and for intellectual production, as national sovereignty and isolationism becomes less and less possible to sustain. The bourgeoisie draws even the most barbaric nations into civilization and compels all nations to adopt its mode of production. It "creates a world after its own image." All become dependent on the bourgeoisie. It has also increased political centralization.

Thus, we see that the means of production and of exchange, which serve as the basis of the bourgeoisie, originated in feudal society. At a certain stage, however, the feudal relations ceased to be compatible with the developing productive forces. Thus the "fetters" of the feudal system had to be "burst asunder," and they were. Free competition replaced the old system, and the bourgeoisie rose to power.

Marx then says that a similar movement underway at the present moment. Modern bourgeois society is in the process of turning on itself. Modern productive forces are revolting against the modern conditions of production. Commercial crises, due, ironically, to over-production, are threatening the existence of bourgeois society. Productive forces are now fettered by bourgeois society, and these crises represent this tension. Yet in attempting to remedy these crises, the bourgeoisie simply cause new and more extensive crises to emerge, and diminish their ability to prevent future ones. Thus, the weapons by which the bourgeoisie overcame feudalism are now being turned on the bourgeoisie themselves.

Commentary

The Communist Manifesto opens with a statement of its purpose, to publicize the views, aims and tendencies of the Communists. As such it is a document intended to be read by the public, and it is meant to be easily grasped by a general audience. It is also meant to be a broad description of what Communism is, both as a theory and as a political movement.

In this first section, Marx already introduces several of the key ideas of his theory. One main idea is that all of history until now is the story of a series of class struggles. Underlying all of history, then, is this fundamental economic theme. The most important concept being discussed here is the concept that each society has a characteristic economic structure. This structure breeds different classes, which are in conflict as they oppress or are oppressed by each other. However, this situation is not permanent. As history "marches" on, eventually the means of production cease to be compatible with the class structure as-is. Instead, the structure begins to impede the development of productive forces. At this point, the existing structure must be destroyed. This explains the emergence of the bourgeoisie out of feudalism. It will also explain the eventual destruction of the bourgeoisie. Marx believes that all of history should be understood in this way--as the process in which classes realign themselves in compliance with changing means of production.

Perhaps the most significant aspect of this theory of history is what it doesnot deem important. In Marx's theory, history is shaped by economic relations alone. Elements such as religion, culture, ideology, and even the individual human being, play a very little role. Rather, history moves according to impersonal forces, and its general direction is inevitable.

Marx believes that this type of history will not go on forever, however. The Manifesto will later argue that the modern class conflict is the final class conflict; the end of this conflict will mark the end of all class relations. This section begins to suggest why this might be, positing some of the ways in which the modern era is unique. First, class antagonisms have been simplified, as two opposing classes, the bourgeoisie and the proletariat, emerge. Secondly, while exploitative relationships were previously hidden behind things like ideology, now the veil has been lifted and everything is seen in terms of self- interest. Thirdly, in order for the bourgeoisie to continue to exist, they must continually revolutionize the instruments of production. This leaves social relations in an unprecedentedly unstable state...

The Communist Manifesto

The Communist Manifesto:

The Communist Manifesto reflects an attempt to explain the goals of Communism, as well as the theory underlying this movement. It argues that class struggles, or the exploitation of one class by another, are the motivating force behind all historical developments. Class relationships are defined by an era's means of production. However, eventually these relationships cease to be compatible with the developing forces of production. At this point, a revolution occurs and a new class emerges as the ruling one. This process represents the "march of history" as driven by larger economic forces. Modern Industrial society in specific is characterized by class conflict between the bourgeoisie and proletariat. However, the productive forces of capitalism are quickly ceasing to be compatible with this exploitative relationship. Thus, the proletariat will lead a revolution. However, this revolution will be of a different character than all previous ones: previous revolutions simply reallocated property in favor of the new ruling class. However, by the nature of their class, the members of the proletariat have no way of appropriating property. Therefore, when they obtain control they will have to destroy all ownership of private property, and classes themselves will disappear.

The Manifesto argues that this development is inevitable, and that capitalism is inherently unstable. The Communists intend to promote this revolution, and will promote the parties and associations that are moving history towards its natural conclusion. They argue that the elimination of social classes cannot come about through reforms or changes in government. Rather, a revolution will be required.

The Communist Manifesto has four sections. In the first section, it discusses the Communists' theory of history and the relationship between proletarians and bourgeoisie. The second section explains the relationship between the Communists and the proletarians. The third section addresses the flaws in other, previous socialist literature. The final section discusses the relationship between the Communists and other parties.

Introduction to Organic Chemistry

Introduction

Chemical reactions involve the making and breaking of bonds. It is essential that we know what bonds are before we can understand any chemical reaction. To understand bonds, we will first describe several of their properties. The bond strength tells us how hard it is to break a bond. Bond lengths give us valuable structural information about the positions of the atomic nuclei. Bond dipoles inform us about the electron distribution around the two bonded atoms. From bond dipoles we may derive electronegativity data useful for predicting the bond dipoles of bonds that may have never been made before.
From these properties of bonds we will see that there are two fundamental types of bonds--covalent and ionic. Covalent bonding represents a situation of about equal sharing of the electrons between nuclei in the bond. Covalent bonds are formed between atoms of approximately equal electronegativity. Because each atom has near equal pull for the electrons in the bond, the electrons are not completely transferred from one atom to another. When the difference in electronegativity between the two atoms in a bond is large, the more electronegative atom can strip an electron off of the less electronegative one to form a negatively charged anion and a positively charged cation. The two ions are held together in an ionic bond because the oppositely charged ions attract each other as described by Coulomb's Law.

Ionic compounds, when in the solid state, can be described as ionic lattices whose shapes are dictated by the need to place oppositely charged ions close to each other and similarly charged ions as far apart as possible. Though there is some structural diversity in ionic compounds, covalent compounds present us with a world of structural possibilities. From simple linear molecules like H2 to complex chains of atoms like butane (CH3CH2CH2CH3), covalent molecules can take on many shapes. To help decide which shape a polyatomic molecule might prefer we will use Valence Shell Electron Pair Repulsion theory (VSEPR). VSEPR states that electrons like to stay as far away from one another as possible to provide the lowest energy (i.e. most stable) structure for any bonding arrangement. In this way, VSEPR is a powerful tool for predicting the geometries of covalent molecules.

The development of quantum mechanics in the 1920's and 1930's has revolutionized our understanding of the chemical bond. It has allowed chemists to advance from the simple picture that covalent and ionic bonding affords to a more complex model based on molecular orbital theory. Molecular orbital theory postulates the existence of a set of molecular orbitals, analogous to atomic orbitals, which are produced by the combination of atomic orbitals on the bonded atoms. From these molecular orbitals we can predict the electron distribution in a bond about the atoms. Molecular orbital theory provides a valuable theoretical complement to the traditional conceptions of ionic and covalent bonding with which we will start our analysis of the chemical bond..

Terms

Anion - A negatively charged ion.

Bond - That which holds together atoms in molecules and ions in lattices.

Cation - A positively charged ion.

Coulomb's Law - A mathematical formula whose consequence is that negatively and positively charged particles attract each other and similarly charged species repel each other.

Covalent Bond - A bond that results from a sharing of electrons between nuclei.

Ion - A charged species created by the gain or loss of an electron from an atom or neutral molecule.

Ionic Bond - A bond that results from electrostatic attraction between oppositely charged ions. The cation is positively charged, while the anion is negatively charged.

Lattice - A regularly repeating three-dimensional array of atoms, molecules, or ions.

Molecular Orbital - A combination of atomic orbitals in molecular orbital theory that provides an orbital description of a molecule analogous to the atomic orbital description of atoms.

Molecular Orbital Theory - A description of bonding that combines atomic orbitals from each bondedatom to produce a set of molecular orbitals.

Molecule - A chemical species containing a covalent bond.

Valence Shell Electron Pair Repulsion Theory - A theory used to predict bonding geometries that states that electron pairs will be distributed about the central atom to minimize electron pair repulsions

Gravitation

Newton and Gravitation

SUMMARY

In 1687 Sir Isaac Newton first published his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) which was a radical treatment of mechanics, establishing the concepts which were to dominate physics for the next two hundred years. Among the book's most important new concepts was Newton's Universal Law of Gravitation. Newton managed to take Kepler's Laws governing the motion of the planets and Galileo's ideas about kinematics and projectile motion and synthesize them into a law which governed both motion on earth and motion in the heavens. This was an achievement of enormous importance for physics; Newton's discoveries meant that the universe was a rational place in which the same principles of nature applied to all objects. The Universal Law of Gravitation has several important features. First, it is an inverse square law, meaning that the strength of the force between two massive objects decreases in proportion to the square of the distance between them as they move farther apart. Second, the direction in which the force acts is always along the line (or vector) connecting the two gravitating objects. Moreover, because there is no "negative mass," gravity is always an attractive force. It is also noteworthy that gravity is a relatively weak force. Modern physicists consider there to be four fundamental forces in nature (the Strong and Weak Nuclear forces, the Electromagnetic force and gravity), of which gravity is the weakest. This means that gravity is only significant when very large masses are being considered.

Terms and Formulae

Terms

Universal Law of Gravitation - Newton's Universal Law of Gravitation states that

where m₁ and m₂ are the masses of any two objects under consideration and r₁ and r₂ are their respective position vectors.

Gravitational Constant - This the G that appears as a constant of proportionality in Newton's Universal Law of Gravitation. It has a value of 6.67×10^-11 Nm²/kg².

Formulae

Equation for the gravitational constant

g =

Newton's Law

Qualitatively Newton's Law of gravitation states that:

Every massive particle attracts every other massive particle with a force directly proportional to the product of their masses and inversely proportional to the square of distance between them

In vector notation, if is the position vector of mass m₁ and is the position vector of mass m₂, then the force on m₁ due to m₂ is given by:

= =

The difference of the two vectors in the numerator gives the direction of the force. The appearance of a cube, instead of a square, in the denominator is in order to cancel this direction-giving factor of | - | in the numerator.

Figure 1.1: Direction of force is the difference of the position vectors.

This force has some remarkable properties. First, we note that it acts at a distance , meaning that irrespective of any intervening matter, every particle in the universe exerts a gravitational force on every other particle. Furthermore, gravity obeys a principle of superposition. This means that to find the gravitational force on any particle it is necessary only to find the vector sum of all the forces from all the particles in the system. For example, the force of the earth on the moon is found by vector summing all the forces between all the particles in the moon and earth. This sounds like an immense task, but actually simplifies calculation.

Gravity as a central force

Newton's Universal Law of Gravitation produces a central force. The force is in the radial direction and depends only on the distance between objects. If one of the masses is at the origin, then () = F(r). That is, the force is a function of the distance between the particles and completely in the direction of . Obviously, the force is also dependent on G and the masses, but these are just constant--the only coordinate on which the force depends is the radial one.

It is easy to show that when a particle is in a central force, angular momentum is conserved, and motion takes place in a plane. First, let us consider the angular momentum:

= (×) = × + × = ×(m) + × = 0

The last equality follows because the cross product of with itself is zero, and since is entirely in the direction of , the cross product of these two vectors is zero also. Since angular momentum does not change over time it is conserved. This is essentially a more general expression of Kepler's Second Law, which we saw (here) also asserted the conservation of angular momentum.

At some time t₀, we have the position vector and velocity vector of the motion that define a plane P with a normal given by = ×. In the previous proof we showed that × does not change in time. This means that = × does not change in time either. Therefore, × = for all t. Since must be orthogonal to , it must always lie in the plane P.

Deducing with Sociological Imagination

Sociology is the scientific study of human groups and social behavior. Sociologists focus primarily on human interactions, including how social relationships influence people's attitudes and how societies form and change. Sociology, therefore, is a discipline of broad scope: Virtually no topic—gender, race, religion, politics, education, health care, drug abuse, pornography, group behavior, conformity—is taboo for sociological examination and interpretation.

Sociologists typically focus their studies on how people and society influence other people, because external, or social, forces shape most personal experiences. These social forces exist in the form of interpersonal relationships among family and friends, as well as among the people encountered in academic, religious, political, economic, and other types of social institutions. In 1959, sociologist C. Wright Mills defined sociological imagination as the ability to see the impact of social forces on individuals' private and public lives. Sociological imagination, then, plays a central role in the sociological perspective.

As an example, consider a depressed individual. You may reasonably assume that a person becomes depressed when something “bad” has happened in his or her life. But you cannot so easily explain depression in all cases. How do you account for depressed people who have not experienced an unpleasant or negative event?

Sociologists look at events from a holistic, or multidimensional, perspective. Using sociological imagination, they examine both personal and social forces when explaining any phenomenon. Another version of this holistic model is the biopsychosocial perspective, which attributes complex sociological phenomena to interacting biological (internal), psychological (internal), and social (external) forces. In the case of depression, chemical imbalances in the brain (biological), negative attitudes (psychological), and an impoverished home environment (social) can all contribute to the problem. The reductionist perspective, which “reduces” complex sociological phenomena to a single “simple” cause, stands in contrast to the holistic perspective. A reductionist may claim that you can treat all cases of depression with medication because all depression comes from chemical imbalances in the brain.

On a topic related to depression, French sociologist Emile Durkheim studied suicide in the late 19th century. Being interested in the differences in rates of suicide across assorted peoples and countries and groups, Durkheim found that social rather than personal influences primarily caused these rates. To explain these differences in rates of suicide, Durkheim examined social integration, or the degree to which people connect to a social group. Interestingly, he found that when social integration is either deficient or excessive, suicide rates tend to be higher. For example, he found that divorced people are more likely to experience poor social integration, and thus are more likely to commit suicide than are married people. As another example, in the past, Hindu widows traditionally committed ritualistic suicide (called “suttee” meaning “good women”) because the cultural pressure at the time to kill themselves overwhelmed them.

Social forces are powerful, and social groups are more than simply the sum of their parts. Social groups have characteristics that come about only when individuals interact. So the sociological perspective and the social imagination help sociologists to explain these social forces and characteristics, as well as to apply their findings to everyday life.

Introduction to Accounting

Introduction to Accounting

Accounting is the language of business. It is the system of recording, summarizing, and analyzing an economic entity's financial transactions. Effectively communicating this information is key to the success of every business. Those who rely on financial information include internal users, such as a company's managers and employees, and external users, such as banks, investors, governmental agencies, financial analysts, and labor unions. These users depend upon data supplied by accountants to answer the following types of questions:

• Is the company profitable?

• Is there enough cash to meet payroll needs?

• How much debt does the company have?

• How does the company's net income compare to its budget?

• What is the balance owed by customers?

• Has the company consistently paid cash dividends?

• How much income does each division generate?

• Should the company invest money to expand?

Accountants must present an organization's financial information in clear, concise reports that help make questions like these easy to answer. The most common accounting reports are called financial statements.

Understanding Financial Statements

The financial statements shown on the next several pages are for a sole proprietorship, which is a business owned by an individual. Corporate financial statements are slightly different. The four basic financial statements are the income statement, statement of owner's equity, balance sheet, and statement of cash flows. The income statement, statement of owner's equity, and statement of cash flows report activity for a specific period of time, usually a month, quarter, or year. The balance sheet reports balances of certain elements at a specific time. All four statements have a three-line heading in the following format.

Income statement. The income statement, which is sometimes called the statement of earnings or statement of operations, is prepared first. It lists revenues and expenses and calculates the company's net income or net loss for a period of time. Net income means total revenues are greater than total expenses. Net loss means total expenses are greater than total revenues. The specific items that appear in financial statements are explained later.

The Greener Landscape Group Income Statement For the Month Ended April 30, 20X2 Revenues Lawn Cutting Revenue $845 Expenses Wages Expense $280 Depreciation Expense 235 Insurance Expense 100 Interest Expense 79 Advertising Expense 35 Gas Expense 30 Supplies Expense 25 Total Expenses 784 Net Income $ 61

Statement of owner's equity. The statement of owner's equity is prepared after the income statement. It shows the beginning and ending owner's equity balances and the items affecting owner's equity during the period. These items include investments, the net income or loss from the income statement, and withdrawals. Because the specific revenue and expense categories that determine net income or loss appear on the income statement, the statement of owner's equity shows only the total net income or loss. Balances enclosed by parentheses are subtracted from unenclosed balances.

The Greener Landscape Group Statement of Owner's Equity For the Month Ended April 30, 20X2 J. Green, Capital, April 1 $ 0 Additions Investments $15,000 Net Income 61 15,061 Withdrawals (50) J. Green, Capital, April 30 $ 15,011

Balance sheet. The balance sheet shows the balance, at a particular time, of each asset, each liability, and owner's equity. It proves that the accounting equation (Assets = Liabilities + Owner's Equity) is in balance. The ending balance on the statement of owner's equity is used to report owner's equity on the balance sheet.

The Greener Landscape Group Balance Sheet April 30, 20X2 ASSETS Current Assets Cash $ 6,355 Accounts Receivable 200 Supplies 25 Prepaid Insurance 1,100 Total Current Assets 7,680 Property, Plant, and Equipment Equipment $18,000 Less: Accumulated Depreciation (235) 17,765 Total Assets $25,445 LIABILITIES AND OWNER'S EQUITY Current Liabilities Accounts Payable $ 50 Wages Payable 80 Interest Payable 79 Unearned Revenue 225 Total Current Liabilities 434 Long-Term Liabilities Notes Payable 10,000 Total Liabilities 10,434 Owner's Equity J. Green, Capital 15,011 Total Liabilities and Owner's Equity $25,445

Statement of cash flows. The statement of cash flows tracks the movement of cash during a specific accounting period. It assigns all cash exchanges to one of three categories—operating, investing, or financing—to calculate the net change in cash and then reconciles the accounting period's beginning and ending cash balances. As its name implies, the statement of cash flows includes items that affect cash. Although not part of the statement's main body, significant non-cash items must also be disclosed.

According to current accounting standards, operating cash flows may be disclosed using either the direct or the indirect method. The direct method simply lists the net cash flow by type of cash receipt and payment category. For purposes of illustration, the direct method appears below.

The Greener Landscape Group Statement of Cash Flows For the Month Ended April 30, 20X2

Cash Flows from Operating Activities
Cash from Customers	$ 870
Cash to Employees	(200)
Cash to Suppliers	(1,265)
Cash Flow Used by Operating Activities	(595)
Cash Flows from Investing Activities
Purchases of Equipment	(8,000)
Cash Flows from Financing Activities
Investment by Owner	15,000
Withdrawal by Owner	(50)
Cash Flow Provided by Financing Activities	14,950
Net Increase in Cash	6,355
Beginning Cash, April 1	0
Ending Cash, April 30	$6,355

Solving Simple Linear Equations

Solving Simple Linear Equations

Algebraic equations are translated from complete English sentences. These equations can be solved. In fact, in order to successfully solve a word problem, an equation must be written and solved.

Look at these two definitions in the following sections and compare the examples to ensure you know the distinction between an expression and an equation.

Defining an Algebraic Expression

An algebraic expression is a collection of constants, variables, symbols of operations, and grouping symbols, as shown in Example 1.

Example 1: 4( x − 3) + 6

Defining an Algebraic Equation

An algebraic equation is a statement that two algebraic expressions are equal, as shown in Example 2.

Example 2: 4( x − 3) + 6 = 14 + 2 x

The easiest way to distinguish a math problem as an equation is to notice an equals sign.

In Example 3, you take the algebraic expression given in Example 1 and simplify it to review the process of simplification. An algebraic expression is simplified by using the distributive property and combining like terms.

Example 3: Simplify the following expression: 4( x − 3) + 6

Here is how you simplify this expression:

1. Remove the parentheses using the distributive property.

   4 x + −12 + 6

2. Combine like terms.

   The simplified expression is 4 x + −6.

Note: This problem does not solve for x. This is because the original problem is an expression, not an equation, and, therefore, cannot be solved.

Four Steps for Solving Simple Linear Equations

In order to solve an equation, follow these steps:

1. Simplify both sides of the equation by using the distributive property and combining like terms, if possible.

2. Move all terms with variables to one side of the equation using the addition property of equations, and then simplify.

3. Move the constants to the other side of the equation using the addition property of equations and simplify.

4. Divide by the coefficient using the multiplication property of equations.

In Example 4, you solve the equation given in Example 2, using the four preceding steps to find the solution to the equation.

Example 4: Solve the following equation: 4( x − 3) + 6 = 14 + 2 x

Use the four steps to solving a linear equation, as follows:

• 1.

  Distribute and combine like terms.

  
  

  
  
• 2a.

  Move all terms with variables to the left side of the equation.

  In this example, add a −2x to each side of the equation.

  
  

  
  

  The addition property of equations states that if the same term is added to both sides of the equation, the equation remains a true statement. The addition property of equations also holds true for subtracting the same term from both sides of the equation.

• 2b.

  Place like terms adjacent to each other and simplify.

  Note: Subtracting 6 is changed to adding −6 because the commutative property of addition works only if all operations are addition.

  
  

  
  
• 3.

  Move the constants to the right side of the equation and simplify.

  
  

  
  

  Note: The opposite operation was used to move the constant.

  
  

  
  
• 4.

  Divide by the coefficient and simplify.

  
  

  
  

  The solution is x = 10.

Example 5: Solve the following equation: 12 + 2(3 x − 7) = 5 x − 4

Use the four steps to solving a linear equation, as follows:

• 1a.

  Distribute and combine like terms.

  
  

  
  
• 1b.

  Place like terms adjacent to each other and simplify.

  
  

  
  
• 2a.

  Move variables to the left side of the equation.

  In this example, add −5 x to each side of the equation.

  
  

  
  
• 2b.

  Place like terms adjacent to each other and simplify.

  Note: All subtractions are changed to addition of a negative number.

  
  

  
  
• 3.

  Move the constants to the right side of the equation and simplify.

  
  

  
  

  Note: The opposite operation was used to move the constant.

• 4.

  Because the coefficient is 1, Step 4 is not necessary.

  The solution is x = −2.

Example 5: Solve the following equation: 6 − 3(2 − x) = −5 x + 40

Use the four steps to solving a linear equation, as follows:

• 1.

  Distribute and combine like terms.

  
  

  
  

  Did you remember to distribute the negative three?

  
  

  
  
• 2a.

  Move variables to the left side of the equation.

  In this example, add 5 x to each side of the equation.

  
  

  
  
• 2b.

  Place like terms adjacent to each other.

  
  

  
  
• 2c.

  Simplify by combining like terms.

  
  

  
  
• 3.

  This step is not necessary in this example because all of the constants are on the right side of the equation.

• 4.

  Divide by the coefficient and simplify.

  
  

  
  

  The solution is x = 5.

Remember: The four steps for solving equations must be done in order, but not all steps are necessary in every problem.