Solving Algebraic Equations

• Basic algebra involves using variables to represent unknown quantities
• Algebraic expressions combine variables, numbers, and operations
• Solving equations involves finding the value(s) of the variable(s) that make the equation true

Solving Equations

• Isolate the variable on one side of the equation
• Use inverse operations to undo the operations performed on the variable
• Addition and subtraction are inverse operations
• Multiplication and division are inverse operations
• Whatever operation is performed on one side of the equation must also be performed on the other side to maintain equality
• Simplify both sides of the equation before isolating the variable
• Combine like terms on each side of the equation
• Distribute any multiplication over parentheses

One-Step Equations

• These equations require only one operation to isolate the variable
• To solve x + 5 = 10, subtract 5 from both sides: x = 5
• To solve 3x = 12, divide both sides by 3: x = 4

Two-Step Equations

• These equations require two operations to isolate the variable
• Follow the order of operations in reverse
• To solve 2x + 3 = 7, first subtract 3 from both sides: 2x = 4
• Then, divide both sides by 2: x = 2

Multi-Step Equations

• These equations require multiple steps to isolate the variable
• Simplify both sides of the equation by combining like terms and distributing
• Use inverse operations to isolate the variable

Equations with Variables on Both Sides

• The goal is to get all the terms with the variable on one side of the equation
• Add or subtract terms to move them to the desired side
• To solve 3x + 2 = x + 6, subtract x from both sides: 2x + 2 = 6
• Then, subtract 2 from both sides: 2x = 4
• Finally, divide both sides by 2: x = 2

Equations with Fractions

• To solve equations with fractions, eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD)
• To solve x/2 + 1/3 = 5/6, the LCD is 6
• Multiply both sides by 6: 3x + 2 = 5
• Then, subtract 2 from both sides: 3x = 3
• Finally, divide both sides by 3: x = 1

Equations with Decimals

• To solve equations with decimals, eliminate the decimals by multiplying both sides of the equation by a power of 10
• Choose the power of 10 such that it moves the decimal point to the right of the last decimal place
• To solve 0.2x + 0.1 = 0.5, multiply both sides by 10: 2x + 1 = 5
• Then, subtract 1 from both sides: 2x = 4
• Finally, divide both sides by 2: x = 2

Expansion

• Expansion, also known as distribution, is a method to remove parentheses from an algebraic expression
• It involves multiplying the term outside the parentheses by each term inside the parentheses

Distributive Property

• The distributive property states that a(b + c) = ab + ac
• For example, 2(x + 3) = 2x + 6
• The distributive property also applies to subtraction: a(b - c) = ab - ac
• For example, 3(y - 2) = 3y - 6

Expanding Binomials

• A binomial is an algebraic expression with two terms
• To expand the product of two binomials, each term in the first binomial is multiplied by each term in the second binomial
• (a + b)(c + d) = ac + ad + bc + bd

FOIL Method

• FOIL is a mnemonic for expanding two binomials: First, Outer, Inner, Last
• First: Multiply the first terms of each binomial
• Outer: Multiply the outer terms of the binomials
• Inner: Multiply the inner terms of the binomials
• Last: Multiply the last terms of each binomial
• To expand (x + 2)(x + 3):
  • First: x * x = x^2
  • Outer: x * 3 = 3x
  • Inner: 2 * x = 2x
  • Last: 2 * 3 = 6
• Then, combine like terms: x^2 + 3x + 2x + 6 = x^2 + 5x + 6

Special Cases

• (a + b)^2 = a^2 + 2ab + b^2
• Example: (x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9
• (a - b)^2 = a^2 - 2ab + b^2
• Example: (y - 4)^2 = y^2 - 2(y)(4) + 4^2 = y^2 - 8y + 16
• (a + b)(a - b) = a^2 - b^2
• Example: (z + 5)(z - 5) = z^2 - 5^2 = z^2 - 25

Expanding Trinomials

• A trinomial is an algebraic expression with three terms
• To expand the product of a binomial and a trinomial, each term in the binomial is multiplied by each term in the trinomial
• For example, (x + 1)(x^2 + 2x + 1) = x(x^2 + 2x + 1) + 1(x^2 + 2x + 1) = x^3 + 2x^2 + x + x^2 + 2x + 1 = x^3 + 3x^2 + 3x + 1

Expanding Complex Expressions

• For more complex expressions, apply the distributive property carefully and systematically
• Combine like terms after each step to simplify the expression
• Use parentheses to keep track of the terms and their signs