To find the equation of a line in standard form, we can start with the point-slope form of a line equation:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. In this case, \((x_1, y_1) = (4, -2)\) and \(m = \frac{1}{2}\).
Plugging in the values, we have:
\[ y - (-2) = \frac{1}{2}(x - 4) \]
This simplifies to:
\[ y + 2 = \frac{1}{2}(x - 4) \]
Next, we can multiply both sides by 2 to eliminate the fraction:
\[ 2(y + 2) = x - 4 \]
Which expands to:
\[ 2y + 4 = x - 4 \]
Now we rearrange this equation to standard form \(Ax + By = C\). To do this, we move \(x\) and the constants to one side:
\[
- x + 2y = -4 - 4 \]
or
\[ -x + 2y = -8 \]
To write it in standard form with \(A\) being positive, we can multiply through by -1:
\[ x - 2y = 8 \]
Thus, the equation of the line in standard form is:
\[ x - 2y = 8 \]
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