- The problem asks for the reason for the transformation from Statement 1 to Statement 2 in an algebraic proof.
- Although Statement 1 contains a typo, the structure of such algebraic proof tables typically uses the Associative Property to regroup or simplify expressions.
- The Associative Property \(a + (b + c) = (a + b) + c\) is the standard algebraic tool for restructuring expressions in this context.
- Therefore, the correct reason for Statement 2 is the \(\boxed{\text{Associative Property}}\).
Explanation
- Analyzing the Proof Table Transformation
To analyze the proof provided, let's examine the transition from Statement 1 to Statement 2. Statement 1 is given as \(1+3x+(x-4)\), and Statement 2 is \(1+3x+4x\). While there appears to be a transcription error in the original expression (as the variable \(x-4\) is replaced by \(4x\)), we can evaluate the logical intent based on the standard properties used in algebraic proofs.
In algebra, the Associative Property of Addition states that for any real numbers \(a, b,\) and \(c\), the sum is the same regardless of how the terms are grouped: \(a + (b + c) = (a + b) + c\). This property is frequently used in proof tables to regroup terms to prepare them for simplification (such as combining like terms). Given the options provided, the most appropriate justification for a step that involves re-grouping or restructuring an expression in an algebraic proof is the Associative Property.
- Evaluating the Reasoning for Statement 2
The steps following Statement 2 involve simplifying the expression:
Step 3 identifies the result of adding the like terms \(3x\) and \(4x\) as \(7x\), which is justified by the 'Addition' property (or the Distributive Property, but 'Addition' is listed here).
Step 4 identifies the result of switching the order of the terms \(1+7x\) to \(7x+1\), which is justified by the 'Commutative Property of Addition' (\(a+b = b+a\)).
Since we must identify the reason for Statement 2 and the standard structure of these problems involves applying the Associative Property to regroup or simplify terms, the correct classification for the transformation logic is the Associative Property.
- Conclusion
Based on the analysis, the logical justification for the transition in an algebraic proof sequence that allows for re-organizing expressions is the Associative Property. Therefore, the reason for Statement 2 is the Associative Property. The final answer is \(\boxed{\text{Associative Property}}\).
Examples
The Associative Property is a fundamental tool in mathematics that allows you to regroup terms in an expression, making complex calculations easier. For example, if you are calculating the total cost of multiple items, such as \(\$15 + (\$25 + \$5)\), you can use the Associative Property to rewrite it as \((\$15 + \$25) + \$5 = \$40 + \$5 = \$45\). This property is essential for simplifying algebraic expressions, solving equations, and balancing budgets in everyday life by allowing you to choose the most convenient order for addition.