Study Notes on Solving Equations

Overview of Solving Equations

  • Solving equations involves finding a value for the variable (e.g., x) that makes the equation true, ensuring that both sides of the equation are equal.

Method 1: Isolating the Variable

  • This method involves rearranging the equation to isolate the variable on one side, allowing us to solve for its value.
  • Example: Solve for x in the equation 2x - 6 = 14.

Steps to Isolate the Variable

  1. Start with the original equation:
    • 2x - 6 = 14
  2. Add 6 to both sides of the equation to eliminate the -6:
    • 2x - 6 + 6 = 14 + 6
    • 2x = 20
  3. Divide both sides by 2 to solve for x:
    • ( x = \frac{20}{2} = 10 )
  4. The solution is found: x = 10.
  5. Check the solution by plugging it back into the original equation:
    • 2(10) - 6 = 14
    • 20 - 6 = 14, confirming that x = 10 is indeed the correct solution.

Method 2: Solving by Factoring

  • Another valid method to solve equations is by factoring, which can be more efficient for certain types of equations.
  • The goal is to rearrange the equation such that one side is equal to zero.

Example: Reapplying the Factoring Method to 2x - 6 = 14

  1. Start with the original equation:
    • 2x - 6 = 14
  2. Subtract 14 from both sides to move all terms to one side:
    • 2x - 6 - 14 = 0
    • 2x - 20 = 0
  3. Factor the equation:
    • Factor out the common term:
    • 2(x - 10) = 0
  4. Set each factor to zero and solve:
    • 2 ≠ 0 is always true
    • x - 10 = 0 leads to x = 10
  5. Thus, we confirm that x = 10 as the solution.

Example 2: Solving x² + 2 = 11

  • Solving this equation using both methods will yield the same results.

Method 1: Isolating the Variable

  1. Start with the equation:
    • x² + 2 = 11
  2. Subtract 2 from both sides:
    • x² = 11 - 2
    • x² = 9
  3. Take the square root of both sides:
    • x = ±√9
    • The solutions are x = 3 and x = -3
  4. Check the solutions:
    • For x = 3:
      • 3² + 2 = 9 + 2 = 11 (True)
    • For x = -3:
      • (-3)² + 2 = 9 + 2 = 11 (True)

Method 2: Solving by Factoring

  1. Start with the rearranged equation:
    • x² + 2 - 11 = 0
    • x² - 9 = 0
  2. Factor the equation:
    • (x + 3)(x - 3) = 0
  3. Set each factor to zero:
    • x + 3 = 0 leads to x = -3
    • x - 3 = 0 leads to x = 3
  4. Confirm all solutions from the isolation method: positive and negative 3 are solutions.

Summary of Methods

  • Both methods, isolating the variable and solving by factoring, yield the same results and are valid approaches for solving equations.
  • Why learn both methods:
    • While isolating variables works well for many equations, there are cases where the factoring method is more efficient or necessary.
    • As we advance in mathematics, understanding multiple solving techniques is critical to tackle more complex problems effectively.

Conclusion

  • In this tutorial, we reviewed two methods for solving equations, specifically with linear and quadratic forms.
  • Going forward, we will explore scenarios where isolation might not be adequate, while factoring may provide a clearer path.