- Identify the slope \(m = -10\) and the point \((2, 7)\).
- Use the point-slope formula: \(y - y_1 = m(x - x_1)\).
- Substitute the known values into the equation to get the final result: \(\boxed{y - 7 = -10(x - 2)}\).
Explanation
- Analyzing the Given Data and Formula
To find the equation of a line in point-slope form, we need to know the slope of the line and a point through which the line passes. The problem provides a slope of \(m = -10\) and a specific point \((x_1, y_1) = (2, 7)\). The point-slope form is a standard way to represent the equation of a linear function and is given by the formula:
\[y - y_1 = m(x - x_1)\]
where \(m\) is the slope and \((x_1, y_1)\) is the given point.
- Formulating the Equation
Now, we substitute the given values into the point-slope formula. We have the slope \(m = -10\), and the coordinates of the point \(x_1 = 2\) and \(y_1 = 7\). Substituting these values into the formula \(y - y_1 = m(x - x_1)\), we obtain:
\[y - 7 = -10(x - 2)\]
This equation represents the line in point-slope form, as required by the problem.
- Finalizing the Equation
The equation \(y - 7 = -10(x - 2)\) is in the requested point-slope form. No further simplification is required as per the instructions to leave it in this specific format. Thus, the final equation is \(\boxed{y - 7 = -10(x - 2)}\).
Examples
In real-life scenarios, the point-slope form is useful when you are observing a trend and want to quickly construct a model. For example, if a company's profit is changing at a steady rate (the slope) and you know their profit on a specific day (the point), you can use the point-slope form to quickly write an equation for the profit at any other time. It serves as a foundation for linear modeling in business, physics, and engineering.