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 2x−2y=−16 2 x minus 2 y equals negative 16 −12x+y=8 negative Start Fraction 1 over 2 End Fraction x plus y equals 8 y=12x+8 y equals Start Fraction 1 over 2 End Fraction x plus 8 x−2y=−16
Answers
GPT-4o mini
To write the equation of the line in standard form, we need to start with the point-slope form of the line equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. In this case, the point is \((2, 9)\) and the slope \(m\) is \(12\).
Plugging these values into the point-slope formula:
\[ y - 9 = 12(x - 2) \]
Now, simplify this equation:
\[ y - 9 = 12x - 24 \]
Now, adding 9 to both sides:
\[ y = 12x - 15 \]
Next, we convert this into standard form (Ax + By = C):
Starting from \(y = 12x - 15\), we subtract \(12x\) from both sides:
\[ -12x + y = -15 \]
To write it in the standard form where \(A\) is positive, we can multiply the entire equation by -1:
\[ 12x - y = 15 \]
Now we can try to match it with the given options. Rearranging this gives:
\[ 12x - y - 15 = 0 \]
However, none of the options presented matches this exactly. We can further check the forms:
- **Option 1**: \(2x - 2y = -16\) \(\Rightarrow x - y = -8\) (Not matching)
- **Option 2**: \(-12x + y = 8\) \(\Rightarrow 12x - y = -8\) (Not matching)
- **Option 3**: \(-\frac{1}{2}x + y = 8\) (Not matching)
- **Option 4**: \(y = 12x + 8\) (Not matching)
- **Option 5**: \(x - 2y = -16\) (Not matching)
None of the provided options match the derived equation \(12x - y = 15\) or \(12x - y - 15 = 0\).
However, let's go through the options given once more, particularly focusing on reformatting or potential inconsistency.
1. **First option**: \(2x - 2y = -16\) rearranges to \(x - y = -8\). Not the same.
2. **Second option**: \(-12x+y=8\) rearranges to \(y = 12x+8\) which is different.
3. **Third option**: Not possible.
4. **Fourth option**: Not possible.
5. **Fifth option**: \(x - 2y = -16\) raises no errors in algebra.
The best outcome centers on either transformation error through numerical matching or input error.
If forced to engage by elimination, revising \(x - 2y = -16\) indicates attributed compromises—considering standard form arrangement limits our visible resolutions.
For complete clarity, the answer is best concluded to focus towards deriving \(12x - y = 15\) as the line equation—properly notably missing in your provided options.
Let me know how else I may assist!
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. In this case, the point is \((2, 9)\) and the slope \(m\) is \(12\).
Plugging these values into the point-slope formula:
\[ y - 9 = 12(x - 2) \]
Now, simplify this equation:
\[ y - 9 = 12x - 24 \]
Now, adding 9 to both sides:
\[ y = 12x - 15 \]
Next, we convert this into standard form (Ax + By = C):
Starting from \(y = 12x - 15\), we subtract \(12x\) from both sides:
\[ -12x + y = -15 \]
To write it in the standard form where \(A\) is positive, we can multiply the entire equation by -1:
\[ 12x - y = 15 \]
Now we can try to match it with the given options. Rearranging this gives:
\[ 12x - y - 15 = 0 \]
However, none of the options presented matches this exactly. We can further check the forms:
- **Option 1**: \(2x - 2y = -16\) \(\Rightarrow x - y = -8\) (Not matching)
- **Option 2**: \(-12x + y = 8\) \(\Rightarrow 12x - y = -8\) (Not matching)
- **Option 3**: \(-\frac{1}{2}x + y = 8\) (Not matching)
- **Option 4**: \(y = 12x + 8\) (Not matching)
- **Option 5**: \(x - 2y = -16\) (Not matching)
None of the provided options match the derived equation \(12x - y = 15\) or \(12x - y - 15 = 0\).
However, let's go through the options given once more, particularly focusing on reformatting or potential inconsistency.
1. **First option**: \(2x - 2y = -16\) rearranges to \(x - y = -8\). Not the same.
2. **Second option**: \(-12x+y=8\) rearranges to \(y = 12x+8\) which is different.
3. **Third option**: Not possible.
4. **Fourth option**: Not possible.
5. **Fifth option**: \(x - 2y = -16\) raises no errors in algebra.
The best outcome centers on either transformation error through numerical matching or input error.
If forced to engage by elimination, revising \(x - 2y = -16\) indicates attributed compromises—considering standard form arrangement limits our visible resolutions.
For complete clarity, the answer is best concluded to focus towards deriving \(12x - y = 15\) as the line equation—properly notably missing in your provided options.
Let me know how else I may assist!