- Identify the slope \(m = -2\) and the point \((4, -2)\).
- Apply the point-slope formula: \(y - y_1 = m(x - x_1)\).
- Substitute the given values to get \(y - (-2) = -2(x - 4)\).
- Simplify the expression to reach the final form: \(\boxed{y + 2 = -2(x - 4)}\).
Explanation
- Analysis of the Given Data
The problem asks for the equation of a line in point-slope form, given its slope and a point it passes through. We are given:
- The slope \(m = -2\).
- A point \((x_1, y_1) = (4, -2)\) through which the line passes.
The point-slope form of a linear equation is defined by the formula:
\[y - y_1 = m(x - x_1)\]
This formula allows us to represent any non-vertical line given its slope and one point on it.
- Substituting and Simplifying the Equation
To find the equation, we substitute the known values \(m = -2\), \(x_1 = 4\), and \(y_1 = -2\) into the point-slope formula:
\[y - (-2) = -2(x - 4)\]
Now, we simplify the expression on the left-hand side by resolving the double negative:
\[y + 2 = -2(x - 4)\]
This equation \(y + 2 = -2(x - 4)\) represents the line in the required point-slope form.
- Conclusion and Final Equation
The resulting equation, \(y + 2 = -2(x - 4)\), correctly uses the slope \(m = -2\) and the given point \((4, -2)\). No further simplification is needed as the goal is specifically the point-slope form. Thus, the equation of the line is \(\boxed{y + 2 = -2(x - 4)}\).
Examples
The point-slope form is highly practical in fields like engineering or economics where you need to model rates of change. For example, if you know the rate at which an item depreciates (the slope, -2 dollars per unit) and its price at a specific quantity (the point), the point-slope form allows you to easily establish the linear function for the item's total cost relative to the number of units sold. Understanding this helps in predicting future values based on current market trends.