Verified Answer :

  • Row 7: \(28x - 14 + \dots\)
  • Row 8: \(125\), \(16\), and \(13x + 14\)
  • Row 9: \(11x - 183\)
  • Row 10: \(18x + 27\), \(22x - 13\), and \(22x - 13\)
  • Row 11: \(4x - 52\) and \(6x - 42\)
  • Row 12: \(42x + 2\) and \(-9x - 42\)

The simplified forms are:
\(\boxed{28x - 14 + \dots, 125, 16, 13x + 14, 11x - 183, 18x + 27, 22x - 13, 22x - 13, 4x - 52, 6x - 42, 42x + 2, -9x - 42}\)

Explanation

  1. Analysis of the Problem Set

To simplify algebraic expressions, we follow the order of operations by distributing coefficients, combining like terms (terms with 'x'), and simplifying constants. Each row represents a set of expressions or problems to be evaluated into their simplest linear form \(ax + b\).

  1. Simplifying Rows 7 and 8

For row 7, we have the expression \(7(4x - 2) + \dots\). Distributing the 7 gives:
\(7 \times 4x - 7 \times 2 = 28x - 14\).
The result is \(28x - 14 + \dots\).
For row 8, we simplify the terms provided:

  1. \(125\) is already a constant.
  2. \(1.6 \times 10 = 16\).
  3. \(4 + 5(3x + 2) - 2x = 4 + 15x + 10 - 2x = (15x - 2x) + (4 + 10) = 13x + 14\).

The row simplifies to \(125\), \(16\), and \(13x + 14\).

  1. Simplifying Rows 9 and 10

For row 9, we simplify \(4(x - 36 - 9) - 3 + 7x\):
\(4(x - 45) - 3 + 7x = 4x - 180 - 3 + 7x = (4x + 7x) - (180 + 3) = 11x - 183\).
For row 10, we evaluate the set of expressions:

  1. \(18x + 27\).
  2. \(9(2x + 3) + 4x - 40 = 18x + 27 + 4x - 40 = 22x - 13\).
  3. \(22x + \dots\) would be \(22x - 13\) if it is intended to match the previous result.

The row simplifies to \(18x + 27\), \(22x - 13\), and \(22x - 13\).

  1. Simplifying Rows 11 and 12

For row 11, we simplify:

  1. \(6(x - 7) - 2(x + 5) = 6x - 42 - 2x - 10 = (6x - 2x) - (42 + 10) = 4x - 52\).
  2. \(6x - 42\).

For row 12, we evaluate:

  1. \(42x + 2\).
  2. \(8 - 4(3x + 9) + 2(x - 7) + x = 8 - 12x - 36 + 2x - 14 + x = (-12x + 2x + x) + (8 - 36 - 14) = -9x - 42\).

The results for rows 11 and 12 are \(4x - 52\), \(6x - 42\) and \(42x + 2\), \(-9x - 42\) respectively.

Examples

Simplifying algebraic expressions is a fundamental skill used in fields like physics and economics. For example, if you are calculating the total cost of a project where the cost is determined by the number of units \(x\) sold, you might encounter complex formulas for revenue and variable costs. By simplifying these into a single linear equation \(ax + b\), you can quickly determine your break-even point or calculate total profit for any given quantity of sales. For instance, if your revenue is \(10(x+5)\) and your costs are \(3x + 20\), you can simplify the profit equation to \(10x + 50 - 3x - 20 = 7x + 30\), which makes it much easier to predict outcomes.

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