What is the Golden Rule for Solving Equations?

The Golden Rule for solving equations is to perform the same operation on both sides of the equation to maintain balance.

Understanding the Golden Rule

The video "0 6 Solving Equations and the Golden Rule - YouTube" at 0:14 and 11:22 explains this concept using the analogy of a balanced scale.

• The Equation as a Balance: Imagine an equation as a perfectly balanced scale. The equals sign (=) represents the fulcrum, or the center of the scale. The expressions on either side of the equals sign are the weights on each side of the scale.
• Maintaining Balance: When you solve an equation, your goal is to isolate the variable (like 'x' or 'y') on one side of the equals sign. To do this, you need to make changes. The Golden Rule states that whatever change you make to one side of the equation, you must make the exact same change to the other side to keep the scale balanced.
• Types of Operations: The video mentions performing the same operation equally on both sides, whether it’s dividing in half, or some "crazy insane operation". These operations include:
  • Addition: Adding the same number to both sides.
  • Subtraction: Subtracting the same number from both sides.
  • Multiplication: Multiplying both sides by the same number.
  • Division: Dividing both sides by the same (non-zero) number.
• Practical Insight: By consistently applying the Golden Rule, you ensure that you are working with equivalent equations. This means that the values of the variables that solve the original equation are the same values that solve the modified equation, ensuring you arrive at the correct answer.

Examples of the Golden Rule in Action

Here are a couple of examples to illustrate the Golden Rule:

1. Example 1: Simple Addition

   • Equation: x - 3 = 7
   • To isolate 'x', we need to add 3 to both sides:
     • x - 3 + 3 = 7 + 3
     • x = 10

2. Example 2: Multiplication and Division

   • Equation: 2y = 12
   • To isolate 'y', we need to divide both sides by 2:
     • 2y / 2 = 12 / 2
     • y = 6

Key Takeaway

The Golden Rule isn't just a mathematical trick; it is a core principle for solving equations. It ensures you are transforming the equation without changing its underlying solution. By consistently applying this rule, you can confidently isolate the variable and find its correct value. Remember, the key is to always maintain balance.