Question

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 \( x \).