Solve for $b$. Express your answer as a proper or improper fraction in simplest terms.
$-\frac{1}{2}-\frac{3}{8} b=-\frac{5}{6}$
The objective is to solve the linear equation \(-\frac{1}{2} - \frac{3}{8}b = -\frac{5}{6}\) for the variable \(b\). Our goal is to isolate \(b\) by performing algebraic operations on both sides of the equation. We will first isolate the term containing \(b\) by adding a constant to both sides, simplify the resulting arithmetic expression, and finally solve for \(b\) by multiplying by the reciprocal of its coefficient.
To begin isolating the term with \(b\), we add \(\frac{1}{2}\) to both sides of the equation:
\[-\frac{3}{8}b = -\frac{5}{6} + \frac{1}{2}\]
Next, we perform the addition on the right side. To add \(-\frac{5}{6}\) and \(\frac{1}{2}\), we find a common denominator, which is \(6\). Since \(\frac{1}{2} = \frac{3}{6}\), the expression becomes:
\[-\frac{3}{8}b = -\frac{5}{6} + \frac{3}{6}\]
\[-\frac{3}{8}b = -\frac{2}{6}\]
Simplifying the fraction \(-\frac{2}{6}\) by dividing the numerator and denominator by their greatest common divisor, \(2\), we get:
\[-\frac{3}{8}b = -\frac{1}{3}\]
Now we solve for \(b\) by multiplying both sides of the equation by the reciprocal of \(-\frac{3}{8}\), which is \(-\frac{8}{3}\):
\[b = \left(-\frac{1}{3}\right) \times \left(-\frac{8}{3}\right)\]
Performing the multiplication:
\[b = \frac{(-1) \times (-8)}{3 \times 3} = \frac{8}{9}\]
The value of \(b\) is \(\frac{8}{9}\). This fraction is in its simplest form as \(8\) and \(9\) share no common factors other than \(1\).
In financial modeling, you might encounter situations where a debt or budget adjustment is represented by a linear equation. For instance, if a company's total adjustment is defined by a constant reduction and a variable portion scaled by a factor, setting this equal to a target outcome helps determine the necessary change. This problem demonstrates the process of isolating the unknown variable, a fundamental skill in solving for target metrics in any quantitative analysis.