- Distribute the slope: \(y - 3 = \frac{1}{2}x - \frac{1}{2}\).
- Add 3 to both sides to solve for \(y\): \(y = \frac{1}{2}x - \frac{1}{2} + 3\).
- Simplify the constant terms: \(y = \frac{1}{2}x + \frac{5}{2}\).
- The correct option is \(\boxed{y=\frac{1}{2} x+\frac{5}{2}}\).
Explanation
- Problem Analysis
The objective is to convert the point-slope form of a linear equation into the slope-intercept form, \(y = mx + b\). The given point-slope equation is \(y - 3 = \frac{1}{2}(x - 1)\). Here, the slope \(m\) is \(\frac{1}{2}\) and the line passes through the point \((1, 3)\). The slope-intercept form, \(y = mx + b\), requires isolating \(y\) on one side of the equation and simplifying the remaining expression.
- Distributing the Slope
To isolate \(y\), we first distribute the slope \(\frac{1}{2}\) to the terms inside the parentheses:
\[y - 3 = \frac{1}{2} \cdot x - \frac{1}{2} \cdot 1\]
\[y - 3 = \frac{1}{2}x - \frac{1}{2}\]
- Isolating y and Simplifying
Next, we add \(3\) to both sides of the equation to isolate \(y\):
\[y = \frac{1}{2}x - \frac{1}{2} + 3\]
To combine the constant terms on the right side, convert \(3\) to a fraction with a denominator of \(2\):
\[3 = \frac{6}{2}\]
Now, combine the terms:
\[y = \frac{1}{2}x - \frac{1}{2} + \frac{6}{2}\]
\[y = \frac{1}{2}x + \frac{5}{2}\]
- Identifying the Correct Option
Comparing our result \(y = \frac{1}{2}x + \frac{5}{2}\) with the given options, we identify that the third option matches our derived equation. Thus, the slope-intercept form of the line is \(y = \frac{1}{2}x + \frac{5}{2}\).
Examples
In real-world applications, such as tracking the growth of a plant or the cost of a utility service, you often have a point where the observation starts and a known rate of change. By converting equations from point-slope to slope-intercept form, you can easily identify the starting value (y-intercept) and predict future values. For example, if a taxi charges a base fee and a rate per mile, this conversion helps you quickly visualize the total cost equation, making it easier to calculate fares for different distances.