Verified Answer :

  • Use the distributive property to expand the term \(4(1-3x)\) to get \(4-12x\).
  • Combine the expanded expression with the rest of the terms: \(4-12x+7x-8\).
  • Group the like terms: \((-12x+7x)\) and \((4-8)\).
  • Calculate the final expression: \(\boxed{-5x-4}\).

Explanation

  1. Analysis of the Expression

The given expression is \(4(1-3x) + 7x - 8\). The objective is to simplify this into a standard linear form, \(Ax + B\). This involves applying the distributive property, grouping like terms, and performing basic arithmetic.

  1. Distribution and Rearrangement

To simplify the first part of the expression, \(4(1-3x)\), we use the distributive property, which states that \(a(b+c) = ab + ac\).
Distributing the \(4\):
\[4 \times 1 - 4 \times 3x = 4 - 12x\]
Now we substitute this back into the original expression:
\[(4 - 12x) + 7x - 8\]

  1. Combining Like Terms and Final Simplification

Next, we group the terms involving \(x\) together and the constant terms together:
\[(-12x + 7x) + (4 - 8)\]
Performing the arithmetic for each group:
\(-12x + 7x = -5x\)
\(4 - 8 = -4\)
Combining these, we get:
\[-5x - 4\]

Examples

Algebraic simplification is a fundamental skill used in many practical fields. For example, if you are calculating the total cost of a project where a fixed fee is charged along with a variable rate based on the number of hours worked (e.g., a plumber's visit), you might have an expression like \(50 + 20(h-1)\), where \(h\) is the number of hours. Simplifying this expression helps you quickly calculate the total charge for any number of hours \(h\) without performing the multi-step calculation every time. Mastering these techniques makes managing budgets and planning resources much faster and more accurate.

Related Questions