Answered

Simplify this expression: $3(x+4.5)+2(2.6 x-3)$


  1. Distribute the 3 and 2 through the parentheses:

$3(x)+3(4.5)+2(2.6 x)-2(3)$


  1. Find each product:

$3 x+13.5+5.2 x-6$


  1. Combine like terms:
A$8.2 x+7.5$
B$15.6 x-19.5$
C$8.2 x+19.5$
D$15.6 x+7.5$

Verified Answer :

  • Apply the distributive property to expand the expression: \(3(x + 4.5) + 2(2.6x - 3) = 3x + 13.5 + 5.2x - 6\).
  • Group and combine the like terms: \((3x + 5.2x) + (13.5 - 6)\).
  • Perform the arithmetic to obtain the simplified result: \(8.2x + 7.5\).
  • Final Answer: \(\boxed{8.2x + 7.5}\).

Explanation

  1. Analysis of the Algebraic Expression

The expression to simplify is \(3(x + 4.5) + 2(2.6x - 3)\). This expression involves algebraic variables and constants, and the goal is to expand the terms using the distributive property and then combine like terms to reach a simplified form. The distributive property states that \(a(b + c) = ab + ac\). We will apply this property to both parenthetical parts of the expression.

  1. Expanding the Expression

We start by applying the distributive property to each part of the expression:
First, for \(3(x + 4.5)\), we distribute the \(3\):
\(3 \times x + 3 \times 4.5 = 3x + 13.5\)
Next, for \(2(2.6x - 3)\), we distribute the \(2\):
\(2 \times 2.6x - 2 \times 3 = 5.2x - 6\)
Now, we combine these results into one expression:
\(3x + 13.5 + 5.2x - 6\)

  1. Combining Like Terms and Final Conclusion

To simplify \(3x + 13.5 + 5.2x - 6\), we group the like terms together. The terms containing \(x\) are \(3x\) and \(5.2x\), and the constant terms are \(13.5\) and \(-6\).
Combining the \(x\)-terms:
\(3x + 5.2x = (3 + 5.2)x = 8.2x\)
Combining the constant terms:
\(13.5 - 6 = 7.5\)
Therefore, the simplified form of the expression is \(8.2x + 7.5\). Comparing this result with the provided options, we see that it matches the first option.

Examples

In real-life scenarios, simplifying expressions like this is essential when calculating costs or budgets. For example, if you are buying items for a school event where the cost of one type of kit is represented by \(3(x + 4.5)\) and another is \(2(2.6x - 3)\), simplifying the expression allows you to quickly find the total cost as a single formula \(8.2x + 7.5\) based on the variable unit price \(x\). This helps in making quick decisions and managing resources efficiently.

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