- The rearrangement of terms in the second step is justified by the Commutative Property of Addition.
- The simplification of \(5x + 2x\) to \(7x\) is justified by the Distributive Property (combining like terms).
- Adding 15 to both sides in the fourth step is justified by the Addition Property of Equality.
- Dividing both sides by 7 in the fifth step is justified by the Division Property of Equality.
- Swapping sides of the equation in the final step is justified by the Symmetric Property of Equality.
- \(\boxed{\text{Commutative, Distributive, Addition, Division, Symmetric}}\)
Explanation
- Analysis of Algebraic Properties
We are given an equation-solving process and need to identify the specific mathematical property that justifies each step listed in the table. Let's analyze the steps sequentially:
- Given: The initial equation is \(76 = 5x - 15 + 2x\).
- Add: Transition from \(76 = 5x - 15 + 2x\) to \(76 = 5x + 2x - 15\). The terms \(-15\) and \(+2x\) have changed their relative positions. This is the Commutative Property of Addition, which states that \(a + b = b + a\).
- Sub: Transition from \(76 = 5x + 2x - 15\) to \(76 = 7x - 15\). Here, \(5x\) and \(2x\) are added together to get \(7x\). This is an application of the Distributive Property (in the form \(ax + bx = (a + b)x\)), often referred to in school as "combining like terms."
- mult: Transition from \(76 = 7x - 15\) to \(91 = 7x\). Here, \(15\) has been added to both sides of the equation. This is the Addition Property of Equality, which states that if \(a = b\), then \(a + c = b + c\). (Note: The table's label "mult" seems inconsistent with the operation performed; we identify the actual property as the Addition Property of Equality).
- Div: Transition from \(91 = 7x\) to \(13 = x\). Both sides are divided by \(7\). This is the Division Property of Equality, which states that if \(a = b\), then \(a/c = b/c\) for \(c
eq 0\).
- Equate: Transition from \(13 = x\) to \(x = 13\). The sides of the equation have been swapped. This is the Symmetric Property of Equality, which states that if \(a = b\), then \(b = a\).
- Summary of Results
Based on the analysis, we have identified the properties justifying each step of the provided equation-solving process:
- Step 2 (Add): The terms were rearranged using the Commutative Property of Addition.
- Step 3 (Sub): Like terms were combined using the Distributive Property.
- Step 4 (mult): The constant term was added to both sides using the Addition Property of Equality.
- Step 5 (Div): Both sides were divided by the coefficient of \(x\) using the Division Property of Equality.
- Step 6 (Equate): The order of the equation was reversed using the Symmetric Property of Equality.
Examples
Understanding algebraic properties is crucial in real-world applications where you need to verify if an equation was solved correctly, such as in computer programming (debugging code that solves for variables), engineering (checking formulas derived for stress or load), or financial modeling (confirming if a balance equation follows proper logic). By identifying these properties, you can systematically audit complex calculations to ensure that every step is logically sound and mathematically valid, building a strong foundation for higher-level mathematics.