Notes on Equivalence in Equations

Equivalence in Equations

Understanding Equivalence

• Equivalence refers to rewriting an equation in different ways while maintaining the same truth value for a given variable (e.g., $x$).
• If one equation is true for a specific $x$, all equivalent equations will also be true for that same $x$, and vice versa.

Example of Equivalence Preserving Operations:

Consider the equation: $3(x + 1) - x = 9$

1. Distribution:
   • Distribute the 3: $3x + 3 - x = 9$
   • This is equivalent to the original equation.
2. Combining Like Terms:
   • Combine $3x$ and $-x$: $2x + 3 = 9$
   • Still equivalent to the original equation.
3. Adding/Subtracting the Same Value:
   • Subtract 3 from both sides: $2x = 6$
   • Maintains equivalence.
4. Multiplying/Dividing by a Non-Zero Constant:
   • Divide both sides by 2: $x = 3$
   • Again, equivalence is preserved.

• These operations (distribution, combining like terms, adding/subtracting the same value from both sides, multiplying/dividing both sides by a non-zero constant) are equivalence-preserving.

Non-Equivalence Preserving Operations

• Performing an operation on only one side of the equation.
  • Starting Equation: $x = 2$
  • Adding 1 to only the left side: $x + 1 = 2$
  • $x = 2$ satisfies the first equation but not the second.
  • Multiplying only the right side by 3: $x = 6$
    • $x = 2$ does not satify $x = 6$

Dividing by a Variable

• Consider equation: $5x = 6x$
• Incorrect Operation: Divide both sides by $x$.
  • This leads to $5 = 6$, which is false and implies no solution.
  • However, this is incorrect because it overlooks the solution $x = 0$.
• Correct Operation: Subtract $5x$ from both sides.
  • $0 = x$, which is equivalent to $5x = 6x$. Both are true when $x = 0$.

Multiplying by Zero

• Multiplying both sides of an equation by zero can lead to non-equivalent statements.
• Starting Equation: $2x = 6$
• Multiply both sides by zero: $0 = 0$
• $0 = 0$ is true for all $x$, but $2x = 6$ is only true for $x = 3$.
• Therefore, they are not equivalent.

Key Takeaways

• Adding or subtracting the same number from both sides preserves equivalence.
• Multiplying or dividing both sides by a non-zero constant preserves equivalence.
• Be cautious when dividing by a variable, especially if it could be zero.
• Multiplying both sides by zero does not preserve equivalence.
• Avoid performing operations on only one side of an equation.