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Extended Addition Property of Equality
The Extended Addition Property of Equality generalizes the basic Addition Property by allowing different but equal quantities to be added to the two sides of an equation, rather than requiring the same quantity on both sides. Formally, for any expressions , , , and :
If and , then .
The basic Addition Property is a special case of this rule (where and happen to be the same value). The extended version is essential to the elimination method for solving systems of equations because it justifies adding the left-hand sides and right-hand sides of two different equations together: each equation is a statement of equality, so the two left sides and two right sides are equal quantities, and combining them preserves equality. For example, given and , the property guarantees that , which simplifies to .
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Extended Addition Property of Equality
A logistics supervisor is balancing two shipping scales that currently show the same weight (a = b). If the supervisor places a 5-pound calibration weight (c) on both scales, which equation represents the Addition Property of Equality?
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In a warehouse inventory system, if Bin A and Bin B contain the same number of items, adding 10 additional items to both bins will ensure they still contain an equal number of items. This principle is known as the Addition Property of Equality.
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A business administrator is adjusting financial records to ensure that balance is maintained across identical accounts. Match the following terms related to the Addition Property of Equality with their correct descriptions or examples.
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A project manager needs to adjust two departmental budgets that are currently identical. Arrange the steps in the correct order to demonstrate the application of the Addition Property of Equality when adding a uniform 'overhead allowance' to both budgets.
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A warehouse supervisor is balancing inventory counts between two identical storage units that currently hold the same number of items. If the supervisor adds 25 additional items to the first unit, what action does the Addition Property of Equality require to ensure the units still contain an equal number of items?
A retail analyst is using the equation x - 20 = 80 to determine the original stock level (x) of a product after 20 units were sold. To isolate the variable x by adding 20 to both sides of the equation, which mathematical property must the analyst recall?
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A logistics coordinator is managing two separate delivery routes. For Route A, the planned distance (d1) is equal to the actual distance traveled (a1). For Route B, the planned distance (d2) is equal to the actual distance traveled (a2). Which mathematical property justifies the statement that the total planned distance (d1 + d2) is equal to the total actual distance (a1 + a2)?
In a business inventory system, if the physical count of Product A equals its recorded value (a = r) and the physical count of Product B equals its recorded value (b = s), the Extended Addition Property of Equality states that the total physical count is equal to the sum of the records, expressed as a + b = ____.
A retail manager is balancing two separate cash registers. For Register 1, the cash on hand (c1) equals the recorded sales (s1). For Register 2, the cash on hand (c2) equals the recorded sales (s2). The Extended Addition Property of Equality states that if c1 = s1 and c2 = s2, then c1 + c2 = s1 + s2.
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An HR payroll specialist is verifying total compensation for an employee. The base salary (S) is equal to the contracted amount (C), and the performance bonus (B) is equal to the approved incentive (I). Match each mathematical expression or term with its correct role in applying the Extended Addition Property of Equality to calculate the total pay.
A project manager is consolidating the financial reports for two different project phases. Phase 1's actual expenses () equal its budget (), and Phase 2's actual expenses () equal its budget (). Arrange the steps below in the correct logical order to demonstrate how the Extended Addition Property of Equality justifies that the total expenses equal the total budget.
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Defining the Extended Addition Property for Financial Auditing
A logistics manager is auditing two regional distribution centers. In Center A, the number of incoming units (Ua) is equal to the number of outgoing units (Oa). In Center B, the number of incoming units (Ub) is equal to the number of outgoing units (Ob). If the manager applies the Extended Addition Property of Equality to find the combined totals of both centers, which equation correctly represents this property?
A project manager is consolidating the financial results of two separate project phases. For Phase 1, the actual cost () is equal to the budget (). For Phase 2, the actual cost () is equal to the budget (). Which statement best describes the principle of the Extended Addition Property of Equality as it applies to the total costs () and the total budget ()?