Choose all answers that apply:
To determine the equivalence of mathematical expressions, we compare the given options to an original expression. In this case, we have two options, A and B, which represent linear expressions in terms of \(g\) and \(h\):
Option A: \((5g + 3h) \cdot 8 = 40g + 24h\)
Option B: \((5g + 3h) \cdot 6 = 30g + 18h\)
Since the original expression that these options are meant to be compared to is not provided, we cannot evaluate which option (if any) is equivalent to it. Equivalence requires that two expressions yield the same numerical value for all possible values of the variables \(g\) and \(h\). Without the target expression, we cannot verify such a relationship.
Given that the target expression is missing, we must evaluate the options based on the information present. Because neither of the provided expressions A or B can be verified against a target, the only logical conclusion within this context is that neither can be confirmed as the correct equivalent. Therefore, option C ("None of the above") is the most appropriate selection.
In algebra, understanding how to expand and simplify expressions is crucial for checking equivalence. For example, if you are calculating the total cost of two different types of items, where \(g\) is the price of a generic item and \(h\) is the price of a hardware piece, an expression like \(8(5g + 3h)\) represents the cost of 8 sets containing 5 generic items and 3 hardware pieces. If you were told the total cost was \(40g + 24h\), you could verify that the expressions are indeed equivalent by distributing the 8: \(8 \times 5g + 8 \times 3h = 40g + 24h\). This skill helps in simplifying complex billing or inventory statements in real life.