To find the equation of a line given a point and a slope, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the given point and \(m\) is the slope. Here, \((x_1, y_1) = (4, -2)\) and \(m = \frac{1}{2}\).
Substituting in these values:
\[ y - (-2) = \frac{1}{2}(x - 4) \]
This simplifies to:
\[ y + 2 = \frac{1}{2}(x - 4) \]
Next, distribute \(\frac{1}{2}\):
\[ y + 2 = \frac{1}{2}x - 2 \]
Now, subtract \(2\) from both sides to isolate \(y\):
\[ y = \frac{1}{2}x - 4 \]
Now we want to convert this equation to standard form, which is generally written as \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers. To convert \(y = \frac{1}{2}x - 4\) into standard form, we can first eliminate the fraction by multiplying every term by \(2\):
\[ 2y = x - 8 \]
Now, we can rearrange this into standard form:
\[ -x + 2y = -8 \]
To make \(A\) positive, we can multiply through by \(-1\):
\[ x - 2y = 8 \]
Thus, the equation of the line in standard form is:
\[ x - 2y = 8 \]