Algebra: Equations

Equations

• An equation is a statement that says two expressions are equal
• It contains an equal sign (=) and is written in the form: ax + by = c, where a, b, and c are constants, and x and y are variables
• Types of equations:
  • Simple equations: have one variable and one solution (e.g., 2x = 6)
  • Linear equations: have one variable and can be written in the form ax + b = c, where a is not equal to zero (e.g., 2x + 3 = 5)
  • Quadratic equations: have one variable and can be written in the form ax^2 + bx + c = 0, where a is not equal to zero (e.g., x^2 + 4x + 4 = 0)
  • Systems of equations: consist of two or more equations with two or more variables (e.g., 2x + 3y = 5 and x - 2y = -3)

Solving Equations

• Addition and subtraction properties:
  • If a = b, then a + c = b + c and a - c = b - c
• Multiplication and division properties:
  • If a = b, then a × c = b × c and a ÷ c = b ÷ c (if c is not equal to zero)
• Solving linear equations:
  • Add or subtract the same value to both sides to isolate the variable
  • Multiply or divide both sides by a coefficient to isolate the variable
• Solving quadratic equations:
  • Factoring: express the equation as a product of binomials and set each factor equal to zero
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Inequalities

• An inequality is a statement that says one expression is greater than, less than, or equal to another expression
• It contains an inequality sign (<, >, ≤, or ≥) and is written in the form: ax + by > c, where a, b, and c are constants, and x and y are variables
• Types of inequalities:
  • Linear inequalities: have one variable and can be written in the form ax + b > c, where a is not equal to zero
  • Compound inequalities: consist of two or more inequalities with the same variable

Solving Inequalities

• Addition and subtraction properties:
  • If a > b, then a + c > b + c and a - c > b - c
• Multiplication and division properties:
  • If a > b, then a × c > b × c if c is positive, and a × c < b × c if c is negative
  • If a > b, then a ÷ c > b ÷ c if c is positive, and a ÷ c < b ÷ c if c is negative
• Solving linear inequalities:
  • Add or subtract the same value to both sides to isolate the variable
  • Multiply or divide both sides by a coefficient to isolate the variable, and flip the inequality sign if the coefficient is negative
• Solving compound inequalities:
  • Solve each inequality separately and combine the solutions using "and" or "or"

Equations

• An equation is a statement that says two expressions are equal, and is written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
• Equations can be classified into simple, linear, quadratic, and systems of equations.
• Simple equations have one variable and one solution, such as 2x = 6.
• Linear equations have one variable and can be written in the form ax + b = c, where a is not equal to zero, such as 2x + 3 = 5.
• Quadratic equations have one variable and can be written in the form ax^2 + bx + c = 0, where a is not equal to zero, such as x^2 + 4x + 4 = 0.
• Systems of equations consist of two or more equations with two or more variables, such as 2x + 3y = 5 and x - 2y = -3.

Solving Equations

• The addition and subtraction properties of equality state that if a = b, then a + c = b + c and a - c = b - c.
• The multiplication and division properties of equality state that if a = b, then a × c = b × c and a ÷ c = b ÷ c (if c is not equal to zero).
• To solve linear equations, add or subtract the same value to both sides to isolate the variable, and multiply or divide both sides by a coefficient to isolate the variable.
• To solve quadratic equations, use factoring or the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a.

Inequalities

• An inequality is a statement that says one expression is greater than, less than, or equal to another expression, and is written in the form ax + by > c, where a, b, and c are constants, and x and y are variables.
• Inequalities can be classified into linear and compound inequalities.
• Linear inequalities have one variable and can be written in the form ax + b > c, where a is not equal to zero.
• Compound inequalities consist of two or more inequalities with the same variable.

Solving Inequalities

• The addition and subtraction properties of inequality state that if a > b, then a + c > b + c and a - c > b - c.
• The multiplication and division properties of inequality state that if a > b, then a × c > b × c if c is positive, and a × c < b × c if c is negative, and if a > b, then a ÷ c > b ÷ c if c is positive, and a ÷ c < b ÷ c if c is negative.
• To solve linear inequalities, add or subtract the same value to both sides to isolate the variable, and multiply or divide both sides by a coefficient to isolate the variable, and flip the inequality sign if the coefficient is negative.
• To solve compound inequalities, solve each inequality separately and combine the solutions using "and" or "or".

American History

Colonial Era (1607-1776)

• Establishment of Thirteen Colonies: British settlers founded colonies along the Eastern seaboard, with Virginia (1607) being the first.
• Pilgrims and Puritans: Pilgrims, seeking religious freedom, founded Plymouth Colony (1620), while Puritans, seeking to reform the Church of England, established the Massachusetts Bay Colony (1629).
• Triangular Trade: This system involved the exchange of slaves, goods, and resources between Europe, Africa, and the Americas, with the Middle Passage being the brutal journey of enslaved Africans to the Americas.
• Proclamation of 1763: The British decree limited westward expansion, leading to colonial discontent, as it prohibited settlement beyond the Appalachian Mountains.

American Revolution (1775-1783)

• Causes of the Revolution: Taxation without representation, restrictions on trade and commerce, and the slogan "No taxation without representation" fueled the rebellion.
• Key Events: The Boston Massacre (1770) marked the first bloodshed, the Boston Tea Party (1773) protested British taxation, and the Declaration of Independence (1776) formally declared American independence.
• American Patriots: Led by George Washington, they fought against British rule, with the Continental Army being formed in 1775.
• Treaty of Paris (1783): Recognized American independence from Great Britain, ending the Revolutionary War.

Westward Expansion and Civil War (1783-1865)

• Manifest Destiny: This belief in the United States' divine right to expand its territory led to westward expansion, fueled by a sense of nationalism and the idea of American exceptionalism.
• Louisiana Purchase (1803): The U.S. acquired approximately 828,000 square miles of land from France, doubling its territory.
• Mexican-American War (1846-1848): The U.S. gained a significant amount of land, including present-day California, Nevada, Utah, Arizona, New Mexico, Texas, and parts of Colorado, Kansas, Oklahoma, and Wyoming.
• American Civil War (1861-1865): The conflict between the Union (North) and Confederacy (South) revolved around slavery and states' rights, with the Emancipation Proclamation (1863) freeing all slaves in Confederate territory.
• Abraham Lincoln: The 16th President issued the Emancipation Proclamation, led the Union to victory, and ultimately abolished slavery with the 13th Amendment (1865).

Industrialization and Imperialism (1865-1914)

• Reconstruction Era (1865-1877): The period aimed to rebuild the South, establish civil rights for freed slaves, and redefine the nation's social and political landscape.
• Industrialization: Rapid growth of industry, technological innovation, and urbanization transformed the U.S. economy, with the development of railroads, steel, and oil industries.
• Robber Barons: Wealthy industrialists, such as John D. Rockefeller and Andrew Carnegie, dominated the economy, often using exploitative practices.
• Spanish-American War (1898): The U.S. emerged as a global power, gaining Puerto Rico, Guam, and the Philippines, and establishing its influence in the Caribbean and Pacific.

World War I and the Roaring Twenties (1914-1929)

• U.S. involvement in World War I (1917-1918): The U.S. entered the war on the Allied side, contributing to the defeat of the Central Powers, and suffering over 116,000 military casualties.
• The Roaring Twenties: Cultural, social, and economic changes defined the decade, with the Jazz Age, Prohibition, and the rise of consumer culture.
• Harlem Renaissance: This cultural and artistic movement among African Americans in the 1920s saw the emergence of notable figures like Langston Hughes, Zora Neale Hurston, and Duke Ellington.