Question

A line passes through the point (2,9)
and has a slope of 12
. Which option correctly writes the equation of the line in standard form?(1 point)
Responses

x−2y=−16
x minus 2 y equals negative 16

y=12x+8
y equals Start Fraction 1 over 2 End Fraction x plus 8

2x−2y=−16
2 x minus 2 y equals negative 16

−12x+y=8

Answer 1

To find the equation of a line in standard form given a point and a slope, we start with the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is the given point \((2, 9)\) and \(m\) is the slope (which is \(12\) in this case). Substituting these values in, we get:

\[
y - 9 = 12(x - 2)
\]

Expanding this equation:

\[
y - 9 = 12x - 24
\]
\[
y = 12x - 15
\]

Now, we can convert this into standard form (Ax + By = C). To do that, we rearrange the equation:

\[
-12x + y = -15
\]

Multiplying through by -1 to make the x-coefficient positive:

\[
12x - y = 15
\]

Now, we can compare this to the options provided:

1. \(x - 2y = -16\)
2. \(y = 12x + 8\)
3. \(2x - 2y = -16\)
4. \(-12x + y = 8\)

None of these options perfectly match the form we've derived \((12x - y = 15)\). However, the first and last options can be checked for equivalence.

If we check \(−12x + y = 8\) (option 4):

Rearranging gives:
\[
y = 12x + 8
\]
which is incorrect based on our derivation.

None of these options match exactly, but the closest is to check for the, normalized if needed:

If we take \(−12x + y = -15\) and simplify noting that if they have direct \frac{-1}{15} that should be reexamined also in context.

Therefore, if there must be an equivalency beyond clearly \( −12x+y = -15\) returned was potentially options misconstrued.

Ultimately option not found.

To suggest checking equivalently perhaps clearer potential typographical errors on their end would be worthwhile.

However, its clear the provided conditions do not yield a precise match and thus indicated further guidance.