- Use the point-slope formula \(y - y_1 = m(x - x_1)\).
- Substitute the given point \((5, 1)\) and slope \(m = \frac{1}{2}\) into the formula.
- The resulting equation is \(y - 1 = \frac{1}{2}(x - 5)\).
- Final Answer: \(\boxed{y - 1 = \frac{1}{2}(x - 5)}\).
Explanation
- Understanding the Point-Slope Formula
The problem asks for the equation of a line that passes through the point \((x_1, y_1) = (5, 1)\) with a slope of \(m = \frac{1}{2}\). To represent this line, we use the point-slope form of a linear equation, which is given by:
\[y - y_1 = m(x - x_1)\]
In this formula, \(m\) represents the slope of the line, and \((x_1, y_1)\) is a specific point that the line passes through. Having identified the given parameters as \(m = \frac{1}{2}\), \(x_1 = 5\), and \(y_1 = 1\), we are ready to substitute these into the standard formula.
- Deriving and Verifying the Equation
To find the equation, we substitute the given coordinates and the slope into the point-slope formula:
\[y - 1 = \frac{1}{2}(x - 5)\]
This equation relates every point \((x, y)\) on the line to the given reference point \((5, 1)\) through the constant slope of \(\frac{1}{2}\). By comparing this result to the provided options, we can identify which equation is correct. The correct equation is clearly \(y - 1 = \frac{1}{2}(x - 5)\).
Examples
The point-slope form is incredibly useful in everyday engineering and construction. For example, if you are designing a wheelchair ramp and know that it needs to pass through a specific point (the start of the ramp at ground level) and maintain a constant slope (like a 1:12 ratio for accessibility), you can use this formula to map out the entire structure. Just as we found the equation of a line from a point and a slope, engineers use these linear models to ensure ramps are safe, functional, and meet accessibility standards.