Verified Answer :

  • Use the distributive property to multiply the coefficient \(- \frac{5}{3}\) by each term inside the parentheses.
  • Calculate the first product: \(-\frac{5}{3} \times 6x = -10x\).
  • Calculate the second product: \(-\frac{5}{3} \times -3 = 5\).
  • Combine the results to get the simplified linear expression: \(\boxed{-10x + 5}\).

Explanation

  1. Analysis of the Expression

We are given the expression \(L = -\frac{5}{3}(6x - 3)\) and we need to simplify it by distributing the constant term across the terms within the parentheses. The expression is a linear function of \(x\), and simplifying it will reveal its structure more clearly.

  1. Distributing the Constant Factor

To simplify \(L = -\frac{5}{3}(6x - 3)\), we apply the distributive property, which states that \(a(b - c) = ab - ac\).

First, we distribute \(-\frac{5}{3}\) to the term \(6x\):
\[-\frac{5}{3} \times 6x = -5 \times \left(\frac{6x}{3}\right) = -5 \times 2x = -10x\]

Next, we distribute \(-\frac{5}{3}\) to the term \(-3\):
\[-\frac{5}{3} \times (-3) = \frac{-5 \times -3}{3} = \frac{15}{3} = 5\]

  1. Final Simplification

Now, we combine the results from the previous step to form the simplified expression for \(L\):
\[L = -10x + 5\]
Thus, the expression \(-\frac{5}{3}(6x - 3)\) simplifies to \(-10x + 5\). The final result is \(\boxed{-10x + 5}\).

Examples

Imagine you are calculating a remaining balance on a store account. Suppose your initial debt is represented by a function of items bought, where the cost of each item is multiplied by a factor related to a discount or a price adjustment. If an adjustment is represented by the formula \(- \frac{5}{3}(6x - 3)\), simplifying it allows you to see the effective rate at which your debt changes (\(-10\)) and the flat adjustment amount (\(5\)) per unit of \(x\). This makes it much easier to track your finances in real-time.

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