Verified Answer :

  • Expand the parentheses on both sides of the equation: \(16m + 6 - 15m - 4 = 9m + 6 - 2m + 2 - 7m\).
  • Simplify both sides by grouping like terms: \(m + 2 = 8\).
  • Isolate the variable \(m\) by subtracting \(2\) from both sides: \(m = 8 - 2 = 6\).
  • The final solution is \(m = \boxed{6}\).

Explanation

  1. Analysis of the equation

We are given the linear equation in the variable \(m\):
\[2(8 m + 3) - 15 m - 4 = 9 m + 6 - 2(m - 1) - 7 m.\]
Our objective is to find the value of \(m\) that satisfies this equation. We will achieve this by expanding the terms within the parentheses, simplifying both sides of the equation by combining like terms, and then isolating the variable \(m\).

  1. Expanding the parentheses

First, we expand the expressions inside the parentheses using the distributive property:

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

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

  1. Combining like terms

Now, we group and combine like terms on each side of the equation:

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

The equation simplifies to:
\[m + 2 = 8.\]

  1. Solving for \(m\)}],example_of_usage:

Finally, we solve for \(m\) by isolating the variable:
\[m = 8 - 2\]
\[m = 6.\]
Thus, the value of \(m\) that satisfies the equation is \(6\).

Examples

Linear equations are fundamental in real-life situations where you need to balance quantities or find an unknown value based on defined relationships. For example, if you are buying supplies, you might have an initial set of items, discounts, and tax costs represented by a linear equation. Solving for the unknown quantity allows you to determine exactly how many items you can afford or what the total price will be before making a purchase. This skill is essential for budgeting and managing resources efficiently.

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