Solve for b. Express your answer as a proper or improper fraction.
$-\frac{5}{8}=-\frac{1}{2}+\frac{3}{4} b$
To solve the equation \(-\frac{5}{8} = -\frac{1}{2} + \frac{3}{4}b\) for \(b\), we first need to isolate the term containing \(b\). This is a linear equation in one variable. Our goal is to manipulate the equation such that \(b\) is by itself on one side, allowing us to determine its value. We will perform algebraic operations step-by-step while maintaining the balance of the equation.
First, we add \(\frac{1}{2}\) to both sides of the equation to isolate \(\frac{3}{4}b\):
\[-\frac{5}{8} + \frac{1}{2} = \frac{3}{4}b\]
To add the fractions on the left, we find a common denominator, which is 8:
\[-\frac{5}{8} + \frac{4}{8} = \frac{3}{4}b\]
Subtracting the fractions, we get:
\[-\frac{1}{8} = \frac{3}{4}b\]
Now, to solve for \(b\), we 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:
\[b = -\frac{1 \times 4}{8 \times 3} = -\frac{4}{24}\]
Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
\[b = -\frac{1}{6}\]
Thus, the value of \(b\) is \(-\frac{1}{6}\).
This type of linear equation is fundamental in algebra for balancing simple systems or determining unknown quantities in real-world scenarios. For example, if you are balancing a budget where \(\frac{3}{4}\) of your hourly wage (\(b\)) minus a fixed cost of \(\frac{1}{2}\) dollar results in a net gain/loss of \(-\frac{5}{8}\) dollars, you can use this calculation to find your actual hourly rate. Mastering these basic algebraic manipulations is the first step toward solving more complex equations in engineering, economics, and physics.