- The equation is given in point-slope form: \(y+7 = \frac{2}{3}(x+6)\).
- Distribute the slope to get \(y+7 = \frac{2}{3}x + 4\).
- Isolate \(y\) by subtracting 7 from both sides, resulting in \(f(x) = \frac{2}{3}x - 3\).
- The correct linear function is \(\boxed{f(x) = \frac{2}{3}x - 3}\).
Explanation
- Problem Analysis
The point-slope form of a linear equation is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. The given equation is \(y+7 = \frac{2}{3}(x+6)\). This can be interpreted as \(y - (-7) = \frac{2}{3}(x - (-6))\), which corresponds to a line with slope \(m = \frac{2}{3}\) passing through the point \((-6, -7)\). We need to express this in the slope-intercept form \(f(x) = mx + b\), where \(y = f(x)\).
- Solving for the Linear Function
To isolate \(y\), we first distribute the slope \(\frac{2}{3}\) to the terms inside the parentheses \((x+6)\):
\[y + 7 = \frac{2}{3} \cdot x + \frac{2}{3} \cdot 6\]
\[y + 7 = \frac{2}{3}x + 4\]
Next, subtract 7 from both sides to solve for \(y\):
\[y = \frac{2}{3}x + 4 - 7\]
\[y = \frac{2}{3}x - 3\]
Thus, the linear function is \(f(x) = \frac{2}{3}x - 3\).
- Conclusion
The calculation shows that the linear function represented by the given equation is \(f(x) = \frac{2}{3}x - 3\). While this specific result does not match the options provided in the prompt, it is the mathematically correct function for the given equation \(y+7 = \frac{2}{3}(x+6)\). Therefore, the linear function is \(\boxed{f(x) = \frac{2}{3}x - 3}\).
Examples
Linear functions are used in many real-life applications. For instance, if you are planning a budget for a project, you can represent your total cost as a linear function of the quantity of items purchased. By identifying the slope (cost per item) and the y-intercept (initial setup cost), you can predict future expenses accurately. Understanding how to convert between different forms of linear equations allows you to easily analyze cost structures and make informed financial decisions.