Algebra: Linear Equations and Expressions

Algebra

• Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations and model relationships.
• Components:
  • Variables (e.g., x, y)
  • Constants (e.g., numbers like 2, 3.5)
  • Operations (addition, subtraction, multiplication, division)
• Expressions vs. Equations:
  • Expressions: Combinations of variables and constants (e.g., 3x + 2)
  • Equations: Statements asserting that two expressions are equal (e.g., 3x + 2 = 11)

Linear Equation in One Variable

• Definition: An equation that can be written in the form ax + b = c, where a, b, and c are constants and x is the variable.
• General Form:
  • ax + b = 0
  • Example: 2x + 3 = 7
• Solutions: The value of x that makes the equation true.
• Steps to Solve:
  1. Isolate the variable term (e.g., move b to the other side).
  2. Divide by a (if a ≠ 0) to solve for x.
  3. Check the solution by substituting back into the original equation.
• Graphical Representation:
  • Represents a straight line on a graph.
  • The solution corresponds to the x-intercept (where the line crosses the x-axis).
• Applications: Used in various fields like science, engineering, economics, and everyday problem-solving.

Algebra

• Branch of mathematics focused on symbols and rules for manipulating them to solve equations and model relationships.
• Includes components such as:
  • Variables: Symbols representing unknown values (e.g., x, y).
  • Constants: Fixed numerical values (e.g., 2, 3.5).
  • Operations: Fundamental mathematical processes (addition, subtraction, multiplication, division).
• Expressions: Combinations of variables and constants (e.g., 3x + 2).
• Equations: Mathematical statements that assert equality between two expressions (e.g., 3x + 2 = 11).

Linear Equation in One Variable

• Defined as an equation that can be structured in the form ( ax + b = c ) with a, b, and c being constants and x a variable.
• General Form: Can also be expressed as ( ax + b = 0 ).
• Example: ( 2x + 3 = 7 ) illustrates a linear equation.
• Solutions: The value of x that satisfies the equation.
• Steps to Solve:
  • Isolate the variable term by moving constant b to the opposite side.
  • Divide by a (where ( a ≠ 0 )) to find the value of x.
  • Verify the solution by substituting it back into the original equation.
• Graphical Representation: Linear equations graph as straight lines.
• The solution corresponds to the point where the line crosses the x-axis (x-intercept).
• Applications: Foundational in fields like science, engineering, and economics; useful for everyday problem-solving.