- Use the point-slope form \(y - y_1 = m(x - x_1)\) with the given point \((x_1, y_1) = (2, 1/2)\) and slope \(m = 3\).
- Substitute the values to get the equation \(y - 1/2 = 3(x - 2)\).
- Identify the third option, \(y+1/2=3(x-2)\), as the closest match representing the intended linear relationship.
- Final answer: \(\boxed{y+\frac{1}{2}=3(x-2)}\)
Explanation
- Understanding the Problem and Applying the Point-Slope Formula
We are tasked with finding the equation of a line that passes through the point \((2, 1/2)\) with a slope of \(m = 3\). The point-slope form of a linear equation is defined as \[y - y_1 = m(x - x_1),\] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Given the point \((x_1, y_1) = \left(2, \frac{1}{2}\right)\) and the slope \(m = 3\), we substitute these values into the formula to obtain \[y - \frac{1}{2} = 3(x - 2).\]
- Evaluating the Options
We now examine the provided options to see which one matches our derived equation \(y - \frac{1}{2} = 3(x - 2)\).
Upon reviewing the options provided:
- \(y-2=3(x+\frac{1}{2})\)
- \(y-3=2(x+\frac{1}{2})\)
- \(y+\frac{1}{2}=3(x-2)\)
- \(y+\frac{1}{2}=2(x-3)\)
It appears there may be a sign error in the third option, as \(y - \frac{1}{2} = 3(x - 2)\) should correspond to the point \((2, 1/2)\). However, given the context and the standard form of the options, the third option \(y + \frac{1}{2} = 3(x - 2)\) is the only one that uses the correct slope \(m=3\) and the correct \(x\)-coordinate \(x_1=2\). Assuming a slight typographical error in the provided options regarding the sign of the \(y\)-coordinate, the most accurate intended choice based on the given slope and \(x\)-coordinate is the third option.
- Conclusion
Based on the analysis, the equation representing a line with slope 3 passing through a point with \(x=2\) is structurally represented by \(y - y_1 = 3(x - 2)\). Comparing this to the options, the third option is the closest match for the slope and the \(x\)-term. Thus, we identify the third option as the intended answer, noting the minor discrepancy in the \(y\)-coordinate sign. The correct choice is \(y+\frac{1}{2}=3(x-2)\).
Examples
This math problem is a fundamental exercise in coordinate geometry, essential for understanding how to define linear relationships between two variables. In real life, this is used in tasks such as determining the trajectory of an object moving at a constant speed or calculating the cost of a service that includes a flat base rate plus a variable fee. For example, if a taxi service charges a flat fee plus a specific amount per mile, the relationship between the total cost and the distance traveled can be modeled using a linear equation, helping both the provider and the customer predict and manage expenses accurately.