Simplifying Algebraic Expressions

• Algebraic expressions are mathematical phrases containing variables, numbers, and operation symbols
• Variable: Letters or symbols represent unknown numbers such as x, y, a, b
• Constant: Numbers with unchanging values, for example, 3, 7, 1/2, 2.5
• Operation symbols indicate mathematical operations like addition (+), subtraction (−), multiplication (× or ⋅), and division (÷ or /)
• Expression example: In 3x + 5, x is the variable, 3 and 5 are constants, and + is the operation symbol

Combining Like Terms

• Like terms share the same variable raised to the same power (e.g., 3x and 5x are like terms, but 3x and 5x² are not)
• Combining like terms simplifies an expression by adding or subtracting like terms' coefficients
• Coefficient: The number multiplied by a variable in an algebraic term, such as 3 in 3x
• Example: To simplify 2x + 3x + 4y + 5y, combine 2x and 3x to get 5x, and 4y and 5y to get 9y, resulting in 5x + 9y
• Only like terms can be combined; in 7a + 3b - 2a, combine 7a and -2a to get 5a + 3b

Understanding Coefficients

• Coefficients are the numerical part of a term that includes a variable, showing how many of that variable are present
• Terms with no visible coefficient have an implied coefficient of 1 (e.g., x is the same as 1x)
  • In 5y, the coefficient = 5
  • In -3a, the coefficient = -3
  • In b, the coefficient = 1
• Understanding coefficients is key for combining like terms and simplifying expressions

Solving Simple Equations

• Equations equate two expressions, using an equals sign (=)
• Solving equations means finding the variable's value that makes the equation true
• Simple equations have one variable and can be solved using basic operations
• Inverse operations isolate the variable
  • Addition and subtraction are inverse operations
  • Multiplication and division are inverse operations

Solving Equations Using Addition or Subtraction

• With addition equations, subtract the same number from both sides to isolate the variable
  • For example: x + 5 = 12 becomes x + 5 - 5 = 12 - 5, simplifying to x = 7
• With subtraction equations, add the same number to both sides to isolate the variable
  • For example: y - 3 = 8 becomes y - 3 + 3 = 8 + 3, simplifying to y = 11
• Always perform the same operation on both sides to maintain equality

Solving Equations Using Multiplication or Division

• With multiplication equations, divide both sides by the variable's coefficient
  • For example: 3a = 15 becomes (3a)/3 = 15/3, simplifying to a = 5
• With division equations, multiply both sides by the number the variable is being divided by
  • For example: b/4 = 6 becomes (b/4) * 4 = 6 * 4, simplifying to b = 24
• Keep the equation balanced by applying operations to both sides

Evaluating Expressions

• Evaluating expressions means finding their value by substituting given variable values and performing operations
• Follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right)
  • For example: Evaluate 2x + 3 when x = 4. Substituting x with 4: 2(4) + 3. Multiply 2 by 4: 8 + 3. Add 8 and 3: 11.

Steps for Evaluating Expressions

• Substitute given values for variables
• Perform operations within parentheses/brackets first
• Evaluate exponents
• Perform multiplication/division from left to right
• Perform addition/subtraction from left to right
  • For example: Evaluate 3(y - 2) + 5 when y = 7
    • Substitute y with 7: 3(7 - 2) + 5
    • Parentheses: 3(5) + 5
    • Multiply: 15 + 5
    • Add: 20

Practical Applications

• Algebraic expressions and equations apply to real-world situations
• For example, calculating costs, determining distances, and solving rate and time problems
• Understanding these concepts aids problem-solving and analytical thinking

Example Problems

• Problem: Apples are $2 and oranges are $3. Write an expression for the cost of x apples and y oranges
  • Solution: 2x + 3y
• Problem: Solve p + 7 = 15
  • Solution: p = 8 (subtract 7 from both sides)
• Problem: Evaluate 4m - 6 when m = 5
  • Solution: 14 (substitute, multiply, then subtract)

Key Concepts Review

• Algebraic expressions: variables, constants, operation symbols
• Combining like terms: add/subtract coefficients of terms sharing a variable and power
• Coefficients: numerical parts of terms with variables
• Solving equations: isolate the variable using inverse operations
• Evaluating expressions: substitute variable values to find the expression's value
• Practice and familiarity are essential for mastering algebra