Verified Answer :

  • Identify the variable terms (\(4x\) and \(2x\)) and constant terms (\(-5\) and \(-7\)).
  • Add the coefficients of the variable terms: \(4 + 2 = 6\).
  • Add the constant terms: \(-5 - 7 = -12\).
  • State the final simplified expression: \(\boxed{6x - 12}\).

Explanation

  1. Analyzing the Expression

We are given the algebraic expression \(4x - 5 + 2x - 7\). Our goal is to simplify this expression by combining like terms. Like terms are those that contain the same variable to the same power, or in the case of constants, those that are pure numbers. Here, we have two types of terms: terms containing the variable \(x\) (which are \(4x\) and \(2x\)) and constant terms (which are \(-5\) and \(-7\)).

  1. Grouping and Combining Like Terms

To simplify the expression, we first group the terms with the variable \(x\) together and the constant terms together. The expression becomes:
\[(4x + 2x) + (-5 - 7)\]
Combining the coefficients of \(x\), we have \(4 + 2 = 6\), so the sum of the terms containing \(x\) is \(6x\). Combining the constant terms, we have \(-5 - 7 = -12\).

  1. Final Simplified Expression

By combining the simplified parts, we obtain the final expression:
\[6x - 12\]
This is the simplest form of the given expression, as there are no further like terms to combine.

Examples

Imagine you are calculating the total cost of a shopping trip where you buy 4 notebooks at a price of \(x\) dollars each, then find a coupon for \(\$5\) off, buy 2 more pens at \(x\) dollars each, and finally get a discount of \(\$7\). The total cost can be represented by the expression \(4x - 5 + 2x - 7\). By simplifying this to \(6x - 12\), you can easily determine the total cost for any given price of \(x\) by just multiplying the price by 6 and subtracting 12.

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