Sure, let's rewrite the given equation [tex]\( y = \frac{1}{6}x - 1 \)[/tex] in the form [tex]\( Ax + By = C \)[/tex] using integer coefficients for [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
1. Start with the given equation:
[tex]\[
y = \frac{1}{6}x - 1
\][/tex]
2. To eliminate the fraction, multiply every term by 6:
[tex]\[
6y = x - 6
\][/tex]
3. Rearrange the terms to achieve the desired form [tex]\( Ax + By = C \)[/tex]:
- Subtract [tex]\( x \)[/tex] from both sides to bring [tex]\( x \)[/tex] to the left-hand side:
[tex]\[
6y - x = -6
\][/tex]
- For standard form, it's conventional to write the [tex]\( x \)[/tex]-term first. We have:
[tex]\[
-x + 6y = -6
\][/tex]
However, it's more standard (and often preferable) to have the coefficient of the [tex]\( x \)[/tex]-term positive. Therefore, multiply through by -1:
[tex]\[
x - 6y = 6
\][/tex]
So, the equation [tex]\( y = \frac{1}{6}x - 1 \)[/tex] in the form [tex]\( Ax + By = C \)[/tex] with integer coefficients is:
[tex]\[
x - 6y = 6
\][/tex]
Where:
- [tex]\( A = 1 \)[/tex]
- [tex]\( B = -6 \)[/tex]
- [tex]\( C = 6 \)[/tex]
Thus, rewritten, it is:
[tex]\[
x - 6y = 6
\][/tex]
