To find the equation of the line in standard form, we start with the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Here, the point is \((2, 9)\) and the slope \(m\) is \(12\).
Substituting the values into the point-slope form:
\[ y - 9 = 12(x - 2) \]
Now, simplifying this equation:
\[ y - 9 = 12x - 24 \] \[ y = 12x - 24 + 9 \] \[ y = 12x - 15 \]
To convert this to standard form \(Ax + By = C\), we'll rearrange it:
\[ -12x + y = -15 \]
To put the equation in standard form with \(A\) being positive, we can multiply the entire equation by \(-1\):
\[ 12x - y = 15 \]
Now, let's rewrite it as:
\[ 12x - y = 15 \]
Now check the provided options for equivalent forms:
The last form corresponds to:
\[ -12x + y = -15 \]
which, after multiplication by \(-1\), gives:
\[ 12x - y = 15 \]
None of the provided options are equivalent to this standard form directly. However, we can adjust our focus to understanding if any of the options match logically.
Looking at the options and matching:
- \(2x - 2y = -16\) (simplifies to \(x - y = -8\), doesn't match).
- \(x - 2y = -16\) (this rearranges to \(x + 16 = 2y\), doesn't match).
- \(-12x + y = 8\) would give the same implications but not consistent with point or slope.
- \(y = 12x + 8\) doesn't match our original slope.
None of the options match the derived standard form of \(12x - y = 15\) or simplified forms directly aligning without remapping.
To summarize, it seems that the equivalent forms based on given points don't respond to listed option equivalences directly. It might be beneficial to verify calculation legality with actual matching closest response checks.
Select appropriately if aligned variants were revised, none strictly match though can identify functional properties/relational correctness otherwise. Please review potential typographical alignments or interpretative overviewes occurring subsequently, as direct line doesn't see resolution within clear options laid out as given.