- Identify the point-slope form equation: \(y - y_1 = m(x - x_1)\).
- Substitute the given point \((5, 2)\) and slope \(m = -2\) into the formula.
- The equation becomes \(y - 2 = -2(x - 5)\).
- The final answer is \(\boxed{y - 2 = -2(x - 5)}\).
Explanation
- Understanding the Point-Slope Form
The point-slope form of a linear equation is a way to express the relationship between coordinates \(x\) and \(y\) for all points on a line, given one specific point \((x_1, y_1)\) and the slope \(m\) of that line. The formula is given by:
\[y - y_1 = m(x - x_1)\]
In this problem, we are given:
- A specific point on the line: \((x_1, y_1) = (5, 2)\)
- The slope of the line: \(m = -2\)
- Substituting the Values into the Formula
To find the equation in point-slope form, we simply substitute the given coordinates and the slope into the formula:
- Replace \(x_1\) with \(5\)
- Replace \(y_1\) with \(2\)
- Replace \(m\) with \(-2\)
Substituting these values into the formula \(y - y_1 = m(x - x_1)\), we get:
\[y - 2 = -2(x - 5)\]
- Conclusion and Final Equation
The equation is now in the required point-slope form. No further simplification is needed as the question specifically asks for the point-slope form using the provided point \((5, 2)\). Thus, the final equation is:
\[\boxed{y - 2 = -2(x - 5)}\]
Examples
The point-slope form is incredibly useful in physics and engineering, for instance, when calculating the trajectory of a projectile or determining the rate of change of a cooling object. If you know the state of a system at one specific time and its rate of change (slope), you can quickly write an equation to predict its state at any other time. This skill allows for efficient modeling of linear systems in everyday life, from predicting travel times to budgeting expenses.