Write a one step equation. Then write two equations that are equivalent to your equation. How can you prove that all three equations are equivalent?

Answer 1

Let's start with a simple one-step equation:

Original Equation:

1. \( x + 5 = 12 \)

Equations Equivalent to the Original: 2. \( x = 12 - 5 \) 3. \( x = 7 \)

Why These Equations Are Equivalent:

To prove that all three equations are equivalent, we can manipulate and derive each equation from the others.

1. From Original Equation to First Equivalent:

   • Start with the original equation: \( x + 5 = 12 \)
   • Subtract 5 from both sides: \[ x + 5 - 5 = 12 - 5 \]
   • Simplifying yields: \[ x = 12 - 5 \]

2. From Original Equation to Second Equivalent:

   • Using the original equation \( x + 5 = 12 \):
   • We already derived \( x = 12 - 5 \).
   • Now, calculate \( 12 - 5 \): \[ 12 - 5 = 7 \]
   • This gives us: \[ x = 7 \]

3. From First Equivalent to Second Equivalent:

   • Starting with \( x = 12 - 5 \):
   • Simplifying the right side gives us: \[ x = 7 \]

Thus, we have shown that we can derive each equation from one another through basic algebraic manipulations, proving that all three equations are equivalent. Each equation represents the same solution for \( x \).