Verified Answer :

  • Expand both sides of the equation using the distributive property.
  • Combine like terms on the left and right sides to simplify the equation to \(m + 2 = 8\).
  • Subtract 2 from both sides to isolate the variable \(m\).
  • The final value for \(m\) is \(\boxed{6}\).

Explanation

  1. Problem Analysis and Strategy

The goal is to solve the linear equation \(2(8m + 3) - 15m - 4 = 9m + 6 - 2(m - 1) - 7m\). We will simplify both sides of the equation by applying the distributive property to remove parentheses and then combining like terms. Finally, we will isolate the variable \(m\) to determine its value.

  1. Expanding the Terms

First, we expand the terms in the parentheses using the distributive property \(a(b+c) = ab + ac\):

  • On the left side, \(2(8m + 3) = 16m + 6\).
  • On the right side, \(-2(m - 1) = -2m + 2\).

Substituting these into the original equation:
\(16m + 6 - 15m - 4 = 9m + 6 - 2m + 2 - 7m\)

  1. Combining Like Terms

Next, we simplify both sides by grouping and combining the like terms:

  • Left side: \((16m - 15m) + (6 - 4) = m + 2\).
  • Right side: \((9m - 2m - 7m) + (6 + 2) = 0m + 8 = 8\).

The equation becomes:
\(m + 2 = 8\)

  1. Solving for m

Now, we isolate the variable \(m\) by subtracting 2 from both sides of the equation:
\(m = 8 - 2\)
\(m = 6\)

  1. Verification and Final Answer

To verify the result, we substitute \(m = 6\) back into the original equation:

  • Left side: \(2(8(6) + 3) - 15(6) - 4 = 2(48 + 3) - 90 - 4 = 2(51) - 94 = 102 - 94 = 8\).
  • Right side: \(9(6) + 6 - 2(6 - 1) - 7(6) = 54 + 6 - 2(5) - 42 = 60 - 10 - 42 = 50 - 42 = 8\).

Since the left side equals the right side, the solution \(m = 6\) is correct. The final answer is \(\boxed{6}\).

Examples

Solving linear equations is fundamental in many real-world situations, such as managing a budget. For example, if you are planning to rent equipment and have two different pricing options—one with a flat fee and an hourly rate, and another with different terms—you can set up an equation to find the exact number of hours that makes both options cost the same. This helps in making informed decisions about the most cost-effective choice for your project.

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