- The addition property of equality states that adding the same value to both sides of an equation preserves the equality (\(a=b \implies a+c=b+c\)).
- In step 1, the equation is \(\frac{2}{3}x - 9 = -13\).
- In step 2, \(9\) is added to both sides of the equation: \(\frac{2}{3}x - 9 + 9 = -13 + 9\).
- Since this transformation directly reflects the addition property of equality, the correct step is step 2.
- The final answer is \(\boxed{\text{step 2}}\).
Explanation
- Analyzing the addition property of equality
The addition property of equality states that for any real numbers \(a\), \(b\), and \(c\), if \(a = b\), then \(a + c = b + c\). This means that adding the same value to both sides of an equation maintains the equality. We need to identify which step in the provided table uses this specific property to solve the equation \(\frac{2}{3}x - 9 = -13\).
- Evaluating the steps
Let us examine the transition from step to step in the provided table:
- Step 1: The original equation is \(\frac{2}{3}x - 9 = -13\).
- Step 2: The equation becomes \(\frac{2}{3}x - 9 + 9 = -13 + 9\). Here, we observe that \(9\) has been added to both sides of the original equation to isolate the term with \(x\).
- Step 3: This simplifies to \(\frac{2}{3}x = -4\).
- Step 4: The equation becomes \(\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot (-4)\). This applies the multiplication property of equality.
- Step 5: The final result is \(x = -6\).
- Conclusion
By comparing our definition of the addition property of equality with the given steps, we can clearly see that in Step 2, the same value (\(9\)) was added to both sides of the equation \(\frac{2}{3}x - 9 = -13\), transforming it into \(\frac{2}{3}x - 9 + 9 = -13 + 9\). Thus, Step 2 is the correct step where this property was applied.
Examples
The addition property of equality is a fundamental tool in solving real-world problems involving balancing values. For example, if you know that two piles of apples have an equal weight, and you add a 5-pound weight to both sides, the piles will remain perfectly balanced. In algebra, this principle allows us to isolate variables by systematically "undoing" arithmetic operations, ensuring that the equality holds throughout every step of the calculation.