Solving Linear Equations: One-variable Equations

Solving Linear Equations: One-variable Equations

• Definition: A linear equation in one variable is an equation that can be expressed in the form ax + b = 0, where:

  • a and b are real numbers
  • x is the variable

• Basic Steps to Solve:

  1. Isolate the variable:
     • Use inverse operations to get x by itself on one side of the equation.
  2. Perform operations on both sides:
     • Whatever you do to one side of the equation, do the same to the other side.
  3. Simplify:
     • Combine like terms and simplify both sides if possible.

• Example: Solve for x in the equation 2x + 3 = 7

  1. Subtract 3 from both sides: 2x = 4
  2. Divide both sides by 2: x = 2

• Types of Solutions:

  • One solution: The equation can be solved to find a unique value for x.
  • No solution: The equation simplifies to a false statement (e.g., 0 = 5).
  • Infinite solutions: The equation simplifies to a true statement for all values of x (e.g., 0 = 0).

• Special Cases:

  • Identity: Equations that are true for all values of x (like 2(x + 1) = 2x + 2).
  • Contradiction: Equations that have no solution (like 3x + 1 = 3x + 2).

• Common Techniques:

  • Combining like terms: Grouping similar terms to simplify.
  • Using the distributive property: To eliminate parentheses (e.g., a(b + c) = ab + ac).
  • Checking solutions: Substitute back into the original equation to verify correctness.

• Applications:

  • Used in various fields including physics, economics, and engineering for solving problems involving direct relationships.

• Graphical Interpretation:

  • A one-variable linear equation can be represented on a number line, and the solution is the point at which the equation holds true.

• Practice Problems:

  • Solve for x: 4x - 5 = 15
  • Solve for x: 3(x - 2) + 4 = 10
  • Determine the nature of the solution: 7x + 2 = 7x - 3

• Key Formulas:

  • General form: ax + b = 0
  • Solution: x = -b/a (if a ≠ 0)

Definition of Linear Equations

• A linear equation in one variable can be expressed as ax + b = 0, where a and b are real numbers and x is the variable.

Steps to Solve Linear Equations

• Isolate the variable: Utilize inverse operations to solve for x.
• Perform operations on both sides: Ensure both sides of the equation remain equal by performing the same operations.
• Simplify: Combine like terms for clearer equations.

Example Solution

• To solve 2x + 3 = 7:
  • Subtract 3: 2x = 4
  • Divide by 2: x = 2

Types of Solutions

• One solution: Unique value for x found.
• No solution: Results in a false statement (e.g., 0 = 5).
• Infinite solutions: Results in a true statement for all x (e.g., 0 = 0).

Special Cases

• Identity: Equations valid for all x (e.g., 2(x + 1) = 2x + 2).
• Contradiction: Equations with no solutions (e.g., 3x + 1 = 3x + 2).

Common Techniques

• Combining like terms: Simplifies equations by grouping similar terms.
• Distributive property: Eliminates parentheses (e.g., a(b + c) = ab + ac).
• Checking solutions: Substitute values back into the original equation to confirm correctness.

Applications

• Linear equations are employed in fields such as physics, economics, and engineering to solve problems involving direct relationships.

Graphical Interpretation

• Represented on a number line; the solution is the point where the equation is true.

Practice Problems

• Solve for x: 4x - 5 = 15
• Solve for x: 3(x - 2) + 4 = 10
• Determine the nature of the solution: 7x + 2 = 7x - 3

Key Formulas

• General form: ax + b = 0
• Solution formula: x = -b/a (provided a ≠ 0)

This quiz covers the basics of solving one-variable linear equations. You'll learn how to isolate the variable, perform operations, and identify the types of solutions possible. Test your understanding with examples and steps provided in the quiz.