- The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\).
- Given the point \((8, -4)\) and the slope \(m = -\frac{5}{6}\), substitute these values into the formula: \(y - (-4) = -\frac{5}{6}(x - 8)\).
- Simplify the expression to get the final equation: \(y + 4 = -\frac{5}{6}(x - 8)\).
- The correct equation is \(\boxed{y + 4 = -\frac{5}{6}(x - 8)}\).
Explanation
- Analysis of the Problem
We are given a point \((x_1, y_1) = (8, -4)\) and a slope \(m = -\frac{5}{6}\). We need to determine the equation of the line passing through this point with this slope in point-slope form. The point-slope form of a linear equation is given by:
\[y - y_1 = m(x - x_1)\]
Here, \(x_1 = 8\), \(y_1 = -4\), and \(m = -\frac{5}{6}\).
- Substituting and Simplifying
Now, we substitute the known values into the point-slope formula:
\[y - (-4) = -\frac{5}{6}(x - 8)\]
Simplifying the left side, we have \(y - (-4) = y + 4\).
Therefore, the equation is:
\[y + 4 = -\frac{5}{6}(x - 8)\]
- Matching with Provided Options
Comparing our derived equation \(y + 4 = -\frac{5}{6}(x - 8)\) with the given options, we identify that it matches the first option:
\[\bigcirc \quad y + 4 = -\frac{5}{6}(x - 8)\]
This is the correct equation of the line in point-slope form.
Examples
Point-slope form is incredibly useful in real-world situations like urban planning or architecture. For instance, if you are designing a ramp for a building entrance, you know the starting point of the ramp and the desired incline (slope). By using the point-slope form \(y - y_1 = m(x - x_1)\), you can quickly determine the elevation \(y\) at any horizontal distance \(x\) along the ramp's path, ensuring it meets safety standards for accessibility.