- Identify the point-slope formula: \(y - y_1 = m(x - x_1)\).
- Substitute the given slope \(m = -\frac{1}{5}\) and the point \((7, 3)\) into the formula.
- The resulting equation is \(\boxed{y - 3 = -\frac{1}{5}(x - 7)}\).
Explanation
- Data and problem analysis
The problem asks for the equation of a line in point-slope form. We are given the slope of the line, \(m = -\frac{1}{5}\), and a specific point through which the line passes, \((x_1, y_1) = (7, 3)\). The point-slope form of a linear equation is a standard way to write the equation of a line when its slope and one point on it are known.
- Applying the point-slope formula
The point-slope form of a linear equation is defined as:
\[y - y_1 = m(x - x_1)\]
where \(m\) is the slope and \((x_1, y_1)\) is the given point on the line. In this problem, we identify the values:
\(m = -\frac{1}{5}\)
\(x_1 = 7\)
\(y_1 = 3\)
By substituting these values directly into the point-slope formula, we can obtain the required equation.
- Conclusion
Substituting the identified values into the formula \(y - y_1 = m(x - x_1)\), we get:
\[y - 3 = -\frac{1}{5}(x - 7)\]
This is the equation of the line in point-slope form, utilizing the given point \((7, 3)\) and the slope of \(-\frac{1}{5}\). Thus, the final equation is \(\boxed{y - 3 = -\frac{1}{5}(x - 7)}\).
Examples
Point-slope form is incredibly useful in real-life engineering and physics, where you often know the rate of change (the slope) and a single observed state of a system (a point). For example, if a car is traveling at a constant rate of -1/5 meters per second relative to a reference point, and you know its position is 7 meters at 3 seconds, the point-slope form allows you to predict its position at any other time. This simple yet powerful equation serves as a foundation for modeling linear relationships in countless fields.