To find the equation of a line in standard form \(Ax + By = C\) given a point and a slope, we can start with the point-slope form of the equation, which is:
\[ y - y_1 = m(x - x_1) \]
Where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. Here, \(m = 12\) and the point is \((2, 9)\).
Substituting these values into the point-slope form:
\[ y - 9 = 12(x - 2) \]
Now, simplify this equation:
\[ y - 9 = 12x - 24 \]
Add 9 to both sides:
\[ y = 12x - 15 \]
Now, to convert this to standard form, we want to rearrange it to \(Ax + By = C\):
\[ -12x + y = -15 \]
Multiplying through by -1 to make the coefficient of \(x\) positive:
\[ 12x - y = 15 \]
Now we need to check if any of the provided options match with \(12x - y = 15\):
- Option 1: \(x - 2y = -16\)
- Option 2: \(y = 12x + 8\)
- Option 3: \(-12x + y = 8\)
- Option 4: \(2x - 2y = -16\)
None of the provided options exactly match \(12x - y = 15\). However, let's check for equivalent forms.
- For Option 1: \(x - 2y = -16\)
Rearranging gives \(x + 16 = 2y\) which is not equivalent.
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For Option 2: \(y = 12x + 8\) is not equivalent.
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For Option 3: \(-12x + y = 8\) also is not equivalent.
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For Option 4: \(2x - 2y = -16\)
Dividing by 2 gives \(x - y = -8\), which is not equivalent.
This indicates that none of the responses correctly write the equation of the line in standard form as derived. The correct rearrangement for standard form from our initial derivation is \(12x - y = 15\).
Could you please double-check the responses given?