- Identify the slope \(m = -\frac{1}{5}\) and the point \((x_1, y_1) = (5, -5)\).
- Use the point-slope formula: \(y - y_1 = m(x - x_1)\).
- Substitute the given values: \(y - (-5) = -\frac{1}{5}(x - 5)\).
- The final equation is \(\boxed{y - (-5) = -\frac{1}{5}(x - 5)}\).
Explanation
- Understanding the Problem and Formula
The problem asks us to find the equation of a line in point-slope form, given a slope \(m = -\frac{1}{5}\) and a point \((x_1, y_1) = (5, -5)\). The point-slope form of a linear equation is defined as \(y - y_1 = m(x - x_1)\). Our objective is to substitute the given values into this formula correctly.
- Substituting and Simplifying Values
We are given the following values:
- Slope: \(m = -\frac{1}{5}\)
- Point: \((x_1, y_1) = (5, -5)\)
Substitute these values into the point-slope formula \(y - y_1 = m(x - x_1)\):
\[y - (-5) = -\frac{1}{5}(x - 5)\]
Simplifying the left side, we have \(y - (-5) = y + 5\). Thus, the equation becomes:
\[y - (-5) = -\frac{1}{5}(x - 5)\]
This is the requested form \(y - \square = \square(x - \square)\). Substituting the values, we get \(y - (-5) = -\frac{1}{5}(x - 5)\).
- Final Equation Formulation
The equation in point-slope form is \(y - (-5) = -\frac{1}{5}(x - 5)\). We have successfully identified all components for the expression, and the values are simplified as required. The final form is \(\boxed{y - (-5) = -\frac{1}{5}(x - 5)}\).
Examples
Point-slope form is a fundamental tool in coordinate geometry, used whenever you know a single point on a line and its steepness. For example, if you are an urban planner designing the slope of a handicap ramp or a road, and you know one specific point the ramp must pass through and the required steepness (slope), the point-slope form allows you to easily determine the entire path of the ramp. It turns descriptive information about a path into a precise mathematical model.