To convert the equation \( y = x + 7 \) to standard form using integers, we need to rearrange it into the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers and \( A \) is non-negative.
Starting with the equation:
\[ y = x + 7 \]
We can rearrange it by subtracting \( x \) from both sides:
\[ -x + y = 7 \]
To express it in standard form, multiply the entire equation by -1:
\[ x - y = -7 \]
Now we have the equation in the standard form \( Ax + By = C \) with \( A = 1 \), \( B = -1 \), and \( C = -7\). However, to ensure that \( A \) is non-negative, we can rewrite it as:
\[ -x + y = 7 \]
This can also be presented as:
\[ -x + 6y = 7 \]
However, it seems that "standard form" was required, so based on this convention:
The standard forms provided in your choices do not match \( Ax + By = C \) form of our equation. However, the closest match preserved in integer format from the rearranging consistent with the ordinary forms shared would be:
\[ -x + 6y = 7 \]
If you are looking for strictly integer forms, then the negative sign can be managed, translating back to:
\[ -1(x - y) = 7 \rightarrow x - y = 7 \]
In case of the need to declare the solutions distinctly without confusion of moving parts:
- The valid equivalent standard form of \( y = x + 7 \) is indeed \( -x + y = 7 \)
- Considering criteria among integers, choose \( -1(x - y) = 7 \) over any others; if looking from provided options for the best fit-solution alignment, \( -x + 6y = 7\) stands out from your preferences shared:
This equation could symbolize a broader context, yet since none is entirely explicit, I've shared the conversion based on algebraic normalization of integer standards.