Solving Linear Equations

Introduction to Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. The solutions to these equations are values of the variable that make the equation true. Mastery of solving linear equations is foundational for all algebraic problem-solving.

• Linear Equation: An equation of the form ax + b = c, where a, b, and c are constants.

• Solution: The value of the variable that satisfies the equation.

Steps for Solving Linear Equations

• Step 1: Simplify both sides of the equation (expand parentheses, combine like terms).

• Step 2: Move all variable terms to one side and constants to the other.

• Step 3: Isolate the variable by performing inverse operations.

• Step 4: Check your solution by substituting it back into the original equation.

Examples of Solving Linear Equations

• Example 1: Solution:

  • Subtract from both sides:

  • Simplify:

  • Subtract $2x = -23$

• Example 2: Solution:

  • Expand:

  • Subtract from both sides:

  • Subtract $14x = -20$

Solving Equations with Fractions

When equations contain fractions, it is often helpful to clear the fractions by multiplying both sides by the least common denominator (LCD).

• Step 1: Identify the LCD of all denominators.

• Step 2: Multiply both sides of the equation by the LCD to eliminate fractions.

• Step 3: Solve the resulting linear equation.

Example:

• Equation: Solution:

  • LCD is 15. Multiply both sides by 15:

  • Subtract from both sides:

  • Add $12

  • Divide by $23x = \frac{12}{23}$

Practice Problems (from the file)

The following problems are typical examples of linear equations and equations with fractions:

Key Concepts and Formulas

• Isolating the Variable: Use inverse operations to get the variable alone on one side of the equation.

• Combining Like Terms: Add or subtract terms with the same variable.

• Clearing Fractions: Multiply both sides by the LCD.

Summary Table: Steps for Solving Linear Equations

Additional info: These notes are based on a worksheet containing practice problems for solving linear equations, including equations with fractions. The content is directly relevant to the "Linear Equations and Inequalities" chapter in Beginning Algebra.