To find the equation of the line in standard form that passes through the points (-4, -1) and (1/2, 3), we first need to determine the slope (m) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let \((x_1, y_1) = (-4, -1)\) and \((x_2, y_2) = \left(\frac{1}{2}, 3\right)\):
\[ m = \frac{3 - (-1)}{\frac{1}{2} - (-4)} = \frac{3 + 1}{\frac{1}{2} + 4} = \frac{4}{\frac{1}{2} + \frac{8}{2}} = \frac{4}{\frac{9}{2}} = 4 \cdot \frac{2}{9} = \frac{8}{9} \]
Now that we have the slope \(m = \frac{8}{9}\), we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point \((-4, -1)\):
\[ y - (-1) = \frac{8}{9}(x - (-4)) \]
This simplifies to:
\[ y + 1 = \frac{8}{9}(x + 4) \] \[ y + 1 = \frac{8}{9}x + \frac{32}{9} \]
Now, subtract 1 (which is \(\frac{9}{9}\)) from both sides:
\[ y = \frac{8}{9}x + \frac{32}{9} - \frac{9}{9} \] \[ y = \frac{8}{9}x + \frac{23}{9} \]
Next, we need to convert this equation into standard form \(Ax + By = C\). We first eliminate the fraction by multiplying the entire equation by 9:
\[ 9y = 8x + 23 \]
Rearranging gives:
\[ -8x + 9y = 23 \]
We can multiply the entire equation by -1 to put it in standard form:
\[ 8x - 9y = -23 \]
Thus, the equation of the line in standard form is:
\[ \boxed{8x - 9y = -23} \]
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