- Recall the point-slope formula: \(y - y_1 = m(x - x_1)\).
- Substitute the given point \((3, -2)\) and slope \(m = -\frac{4}{5}\) into the formula.
- Simplify the expression: \(y - (-2) = -\frac{4}{5}(x - 3)\) becomes \(y + 2 = -\frac{4}{5}(x - 3)\).
- Identify the matching option, which is \(\boxed{y + 2 = -\frac{4}{5}(x - 3)}\).
Explanation
- Analysis of the Problem
The problem asks to find the correct point-slope form equation for a line that passes through the specific point \((x_1, y_1) = (3, -2)\) and has a slope of \(m = -\frac{4}{5}\). The point-slope form of a linear equation is a standard method to express the relationship between coordinates of points on a line given its slope. The general formula is defined as \(y - y_1 = m(x - x_1)\), where \(m\) represents the slope and \((x_1, y_1)\) represents a point that lies on the line.
- Substitution and Simplification
To determine the correct equation, we substitute the known values into the point-slope formula \(y - y_1 = m(x - x_1)\). Here, our slope is \(m = -\frac{4}{5}\), and the coordinates of our point are \(x_1 = 3\) and \(y_1 = -2\). Substituting these values into the formula gives:
\[y - (-2) = -\frac{4}{5}(x - 3)\]
Subtracting a negative value is equivalent to addition, so the term \(y - (-2)\) simplifies to \(y + 2\). The resulting equation is:
\[y + 2 = -\frac{4}{5}(x - 3)\]
- Selection of the Correct Option
Comparing our result \(y + 2 = -\frac{4}{5}(x - 3)\) with the provided options:
- \(y - 3 = -\frac{4}{5}(x + 2)\)
- \(y - 2 = \frac{4}{5}(x - 3)\)
- \(y + 2 = -\frac{4}{5}(x - 3)\)
- \(y + 3 = \frac{4}{5}(x + 2)\)
The third option matches our derived equation exactly. Thus, the correct point-slope equation is \(y + 2 = -\frac{4}{5}(x - 3)\).
Examples
In real life, the point-slope form is often used in fields like physics or engineering to describe the trajectory of an object. For example, if a drone is flying at a constant angle (the slope) and passes through a known GPS coordinate (the point), you can use the point-slope formula to predict its path at any future time. By writing the line equation as \(y - y_1 = m(x - x_1)\), you can easily determine the altitude (\(y\)) for any given horizontal distance (\(x\)) from the starting position.