Verified Answer :

  • Add \(m\) to both sides: \(d + m = \frac{2}{5}b\).
  • Multiply by \(\frac{5}{2}\) to isolate \(b\): \(b = \frac{5}{2}(d + m)\).
  • The result is \(b = \frac{5}{2}d + \frac{5}{2}m\).
  • Final answer: \(\boxed{\frac{5}{2}(d + m)}\).

Explanation

  1. Analysis of the Problem

The objective is to solve the given linear equation for the variable \(b\). The equation is given as \(d = \frac{2}{5}b - m\), where \(d\), \(b\), and \(m\) are real numbers. To find \(b\), we need to perform algebraic manipulations to isolate it on one side of the equation.

  1. Solving for b

Start with the equation:
\[d = \frac{2}{5}b - m\]
To isolate the term containing \(b\), we add \(m\) to both sides:
\[d + m = \frac{2}{5}b\]
Now, to solve for \(b\), we multiply both sides of the equation by the reciprocal of the coefficient \(\frac{2}{5}\), which is \(\frac{5}{2}\):
\[b = \frac{5}{2}(d + m)\]
Distributing the \(\frac{5}{2}\) gives:
\[b = \frac{5}{2}d + \frac{5}{2}m\]
Thus, the solution for \(b\) in terms of \(d\) and \(m\) is \(b = \frac{5}{2}(d + m)\).

Examples

In many real-world scenarios, such as calculating the base amount from a discounted price, we rearrange formulas to solve for a specific variable. For instance, if a tax \(m\) was subtracted from a total amount \(b\) scaled by a factor of \(\frac{2}{5}\) to result in a final price \(d\), we use this algebraic manipulation to find the original base amount \(b\). This is fundamental in accounting, engineering, and various physical science calculations where variables are related by linear equations.

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