Algebra: Equations

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the general form of a linear equation with one variable?

ax + b = c (correct)

ax - b = c

ax + by = c

ax^2 + bx + c = 0

Which of the following is an example of a quadratic equation?

2x - 3y = 5

x - 2y = -3

2x + 3 = 5

x^2 + 4x + 4 = 0 (correct)

What is the purpose of the multiplication and division properties in solving equations?

To check if the solution is correct

To add or subtract the same value to both sides

To isolate the variable by multiplying or dividing both sides by a coefficient (correct)

To simplify the equation by combining like terms

What is the difference between a linear equation and a linear inequality?

A linear equation has an equal sign, while a linear inequality has an inequality sign Signup and view all the answers

How do you solve a compound inequality?

Solve each inequality separately and combine the solutions using 'and' Signup and view all the answers

What is the quadratic formula used for?

Solving quadratic equations Signup and view all the answers

What is the purpose of factoring in solving quadratic equations?

To express the equation as a product of binomials and set each factor equal to zero Signup and view all the answers

What happens to the inequality sign when you multiply or divide both sides of a linear inequality by a negative coefficient?

The sign changes to the opposite inequality sign Signup and view all the answers

What was the main reason behind the Proclamation of 1763?

To limit westward expansion and prevent conflict with Native Americans Signup and view all the answers

What was the main goal of the American Patriots during the American Revolution?

To gain independence from Great Britain Signup and view all the answers

What was the outcome of the Mexican-American War?

The United States gained a significant amount of land, including present-day California and Texas Signup and view all the answers

What was the main issue that led to the American Civil War?

States' rights and slavery Signup and view all the answers

What was the main goal of the Reconstruction Era?

To rebuild the South and establish civil rights for freed slaves Signup and view all the answers

What was the significance of the Louisiana Purchase?

It doubled U.S. territory through land acquisition from France Signup and view all the answers

What was the main characteristic of the Robber Barons?

They were wealthy industrialists who dominated the economy Signup and view all the answers

What was the main outcome of the Spanish-American War?

The United States gained control of the Philippines Signup and view all the answers

Study Notes

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, 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, 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.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Understand the concept of equations, including types of equations such as simple, linear, and quadratic equations. Learn how to write and solve equations in the form ax + by = c.