- Identify the given point \((x_1, y_1) = (8, -4)\) and the slope \(m = -\frac{5}{6}\).
- Use the point-slope formula \(y - y_1 = m(x - x_1)\) to substitute the values.
- Simplify the expression: \(y - (-4) = -\frac{5}{6}(x - 8)\) leads to \(y + 4 = -\frac{5}{6}(x - 8)\).
- Conclude that the correct equation is \(\boxed{y + 4 = -\frac{5}{6}(x - 8)}\).
Explanation
- Understanding the Problem
We are tasked with finding the equation of a line that passes through the specific point \((8, -4)\) and has a constant slope of \(m = -\frac{5}{6}\). The point-slope form of a linear equation is a fundamental tool for representing lines when we know a point on the line and its steepness. This form is expressed as:
\[y - y_1 = m(x - x_1)\]
where \((x_1, y_1)\) is the given point and \(m\) is the slope of the line.
- Applying the Formula
To find the required equation, we identify the given values:
- The point \((x_1, y_1) = (8, -4)\)
- The slope \(m = -\frac{5}{6}\)
Now, we substitute these values into the point-slope formula:
\[y - (-4) = -\frac{5}{6}(x - 8)\]
Simplifying the term on the left side, \(y - (-4)\) becomes \(y + 4\). Thus, the equation becomes:
\[y + 4 = -\frac{5}{6}(x - 8)\]
- Final Result
Comparing our derived equation to the given options:
- \(y + 4 = -\frac{5}{6}(x - 8)\)
- \(y - 4 = -\frac{5}{6}(x + 8)\)
- \(y - 4 = -\frac{5}{6}(x - 8)\)
- \(y + 4 = -\frac{5}{6}(x + 8)\)
Our result matches the first option. Therefore, the correct equation for the line in point-slope form is:
\[\boxed{y + 4 = -\frac{5}{6}(x - 8)}\]
Examples
Point-slope form is widely used in fields like physics and engineering to model relationships between two variables when you know a starting condition and a rate of change. For example, if you are tracking the temperature of a cooling object over time, knowing the temperature at a specific minute and how fast it is decreasing allows you to set up a linear model to predict temperatures at other times using this exact formula. It provides a simple way to define a linear path in a coordinate system based on known data points.