Verified Answer :

  • Add \(\frac{1}{2}\) to both sides to get \(-\frac{1}{8} = \frac{3}{4}b\).
  • Multiply both sides by the reciprocal \(\frac{4}{3}\) to isolate \(b\).
  • Simplify the resulting fraction \(-\frac{4}{24}\) to obtain \(b = -\frac{1}{6}\).
  • The final answer is \(\boxed{-\frac{1}{6}}\).

Explanation

  1. Understanding the Problem

The goal is to solve the linear equation \(-\frac{5}{8} = -\frac{1}{2} + \frac{3}{4}b\) for the variable \(b\). To isolate \(b\), we need to perform algebraic operations, starting by moving the constant term to the other side and then isolating \(b\) by multiplying by the reciprocal of its coefficient.

  1. Isolating the Variable Term

First, add \(\frac{1}{2}\) to both sides of the equation to isolate the term containing \(b\):
\[-\frac{5}{8} + \frac{1}{2} = \frac{3}{4}b\]
To add the fractions on the left, we find a common denominator, which is 8. Since \(\frac{1}{2} = \frac{4}{8}\), we have:
\[-\frac{5}{8} + \frac{4}{8} = \frac{3}{4}b\]
\[-\frac{1}{8} = \frac{3}{4}b\]

  1. Solving for \(b\) and Simplifying

To solve for \(b\), multiply both sides of the equation by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\):
\[b = -\frac{1}{8} \times \frac{4}{3}\]
Performing the multiplication:
\[b = -\frac{1 \times 4}{8 \times 3} = -\frac{4}{24}\]
Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, 4:
\[b = -\frac{1}{6}\]

Examples

Linear equations are used in various real-life scenarios, such as budgeting. Suppose you are adjusting your monthly savings: if your initial balance is \(\$ -\frac{5}{8}\) thousand (representing a debt or specific account adjustment) and you want to reach a target value of \(\$ -\frac{1}{2}\) thousand by adding a variable contribution of \(\frac{3}{4}b\) (where \(b\) is a multiplier factor), you would solve this exact equation to determine the necessary value of \(b\). Mastering this algebraic manipulation allows for precise financial and resource planning.

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