Solving Linear Equations Lecture Notes

Introduction

• Objective: Practice solving linear equations with 7 examples.
• Approach:
  • Pause video and try examples on your own.
  • Examples are time-stamped for convenience.
  • Examples range in difficulty.

General Strategy for Solving Linear Equations

• Definition: Solving a linear equation involves finding all solutions that make the equation true.
• Solutions:
  • One solution
  • Infinitely many solutions
  • No solution
• Goal: Isolate variable (e.g., x = number).
• Steps:
  1. Use properties like distributive, commutative, associative.
  2. Combine like terms (e.g., 3x - 2x = x).
  3. Perform arithmetic operations on both sides.

Example 1

• Equation: 3x - 17 = 19
• Strategy:
  • Add 17 to both sides to isolate 3x.
  • Divide by 3 to solve for x.
• Solution: x = 12
• Solution Set: {12}

Example 2

• Equation with Parentheses: 3(x - 6) = 2x - 5
• Strategy:
  • Use distributive property: Expand left side.
  • Subtract 2x and solve for x.
• Solution: x = 13
• Solution Set: {13}

Example 3

• Potential Contradiction: 3x - 10 = 3x + 7
• Observation:
  • Subtract 3x on both sides results in -10 = 7 (False)
• Conclusion: No solutions.
• Solution Set: Empty set (Ø)

Example 4

• Infinite Solutions: 7(p + 2) - 4p = 3p + 14
• Observation:
  • Simplifies to 3p + 14 = 3p + 14
  • True for all real numbers.
• Solution Set: (-∞, ∞)

Case Summary

• One Solution: x = number
• Infinite Solutions: Simplifies to a true equation (e.g., 0 = 0)
• No Solution: Simplifies to a false equation (e.g., 0 = 2)

Example 5

• Equation with Fractions: (1/5)(n - 5) = 3/5n + 1/10
• Strategy:
  • Multiply by least common denominator to clear fractions.
  • Simplify and solve for n.
• Solution: n = -4
• Solution Set: {-4}

Example 6

• Complex Equation: (-2)[5 - 2(z + 1)] - 4 = 2(3 - z)
• Strategy:
  • Solve inside out (brackets and parentheses first).
  • Use distributive property and combine terms.
• Solution: z = 3
• Solution Set: {3}

Example 7

• Final Example with Fractions
• Strategy:
  • Clear fractions using least common denominator.
  • Distribute, combine terms, and solve.
• Solution: t = 3/2
• Solution Set: {3/2}

Closing Tips

• Check solutions by substituting back into the original equation.
• Ensure the result is a true statement to verify correctness.