- The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope.
- Comparing \(y - 3 = -\frac{1}{2}(x - 2)\) to this form reveals that the slope \(m\) is \(-\frac{1}{2}\).
- The correct choice is \(-\frac{1}{2}\).
- The final answer is \(\boxed{-\frac{1}{2}}\).
Explanation
- Identifying the Slope in Point-Slope Form
The equation provided is \(y - 3 = -\frac{1}{2}(x - 2)\). This equation is in the point-slope form, which is represented by the formula \(y - y_1 = m(x - x_1)\). In this form, the variable \(m\) represents the slope of the line, while \((x_1, y_1)\) represents a specific point through which the line passes. By comparing the given equation to the general form, we can see that \(m = -\frac{1}{2}\), \(x_1 = 2\), and \(y_1 = 3\). Thus, the slope of the line is \(-\frac{1}{2}\).
Examples
Understanding the slope of a line is essential in various fields, such as economics, where it can represent the rate of change of demand for a product as its price changes. For example, if a company models its sales using a linear equation, the slope indicates how many additional units they expect to sell for every one-dollar reduction in price. Mastering this concept helps in making informed decisions about pricing, growth projections, and resource allocation.