- Combining \(\frac{4}{5} + 5\) is a valid first step due to the order of operations.
- The property \(A - B = A + (-B)\) confirms that subtracting \(3x\) is the same as adding \(-3x\).
- The distributive property \(a(b - c) = ab - ac\) confirms that \(\frac{2}{3}\) multiplies both \(14x\) and \(-3\).
- These three statements accurately describe the simplification process of the expression.
Explanation
- Analysis of the Expression and Principles
We are analyzing the algebraic expression \(\frac{2}{3}(14x - 3) - 3x\left(\frac{4}{5} + 5\right)\). To simplify this expression correctly, we need to apply standard algebraic properties such as the distributive property, order of operations (PEMDAS/BODMAS), and the relationship between subtraction and the addition of the opposite. We will evaluate each of the five given statements to identify which ones are true.
- Evaluating Statement 1
Statement 1: "Combining \(\frac{4}{5} + 5\) can be done before multiplying it by \(3x\)."
According to the order of operations, operations within parentheses should be performed first. In the term \(-3x\left(\frac{4}{5} + 5\right)\), we can simplify the expression inside the parentheses: \(\frac{4}{5} + 5 = \frac{4}{5} + \frac{25}{5} = \frac{29}{5}\). Thus, we could rewrite the term as \(-3x \cdot \left(\frac{29}{5}\right)\). This is a valid algebraic step. Therefore, this statement is true.
- Evaluating Statement 2
Statement 2: "Using the distributive property, \(3x\) times \(\frac{4}{5}\) is subtracted, but the \(3x\) times \(5\) is added."
The expression is \(-3x\left(\frac{4}{5} + 5\right)\). Distributing the \(-3x\) gives \(-3x \cdot \frac{4}{5} - 3x \cdot 5\). Both terms involve subtraction (or the addition of a negative value). The claim that the second part is "added" is incorrect because of the leading minus sign before the \(-3x\). Therefore, this statement is false.
- Evaluating Statement 3
Statement 3: "Combining \(14x - 3\) can be done before multiplying by \(\frac{2}{3}\)."
The expression inside the parentheses is \(14x - 3\). These are unlike terms—one involves a variable \(x\), and the other is a constant. They cannot be combined into a single numeric value. Therefore, this statement is false.
- Evaluating Statement 4
Statement 4: "Subtracting \(3x\) is the same as adding \(-3x\)."
This is a fundamental rule in algebra which states that subtracting a value \(B\) is equivalent to adding the additive inverse of \(B\), specifically \(A - B = A + (-B)\). Here, subtracting \(3x\) is exactly the same as adding \(-3x\). Therefore, this statement is true.
- Evaluating Statement 5
Statement 5: "Using the distributive property, the \(\frac{2}{3}\) would be multiplied to \(14x\) and to \(-3\)."
The distributive property states that \(a(b - c) = a \cdot b - a \cdot c\). Applying this to the term \(\frac{2}{3}(14x - 3)\), we multiply \(\frac{2}{3}\) by \(14x\) and \(\frac{2}{3}\) by \(-3\), resulting in \(\left(\frac{2}{3} \cdot 14x\right) - \left(\frac{2}{3} \cdot 3\right)\). This correctly describes the process. Therefore, this statement is true.
- Conclusion
Based on our analysis, the three true statements are:
- Combining \(\frac{4}{5} + 5\) can be done before multiplying it by \(3x\).
- Subtracting \(3x\) is the same as adding \(-3x\).
- Using the distributive property, the \(\frac{2}{3}\) would be multiplied to \(14x\) and to \(-3\).
The correct choice is the set containing these three true statements.
Examples
In fields like structural engineering or financial planning, managing complex formulas is common. For example, when calculating the total cost of a construction project involving multiple materials, you might have an expression like \(5(cost_A - overhead) - 2 \cdot quantity(material_B + material_C)\). Knowing how to distribute terms correctly or simplify constants before multiplication helps ensure accuracy in budget estimations and resource management. Understanding these algebraic rules prevents errors that could lead to significant financial discrepancies.