#### Answer

$\text{slope-intercept form: }
y=x-2
\\\\
\text{standard form: }
x-y=2$

#### Work Step by Step

Using $y-y_1=m(x-x_1)$ or the point-slope form of linear equations, the equation of the line passing through $(
12,10
)$ and with a slope of $m=
1
$ is
\begin{array}{l}\require{cancel}
y-10=1(x-12)
.\end{array}
In the form $y=mx+b$, the equation above is equivalent to
\begin{array}{l}\require{cancel}
y-10=1(x-12)
\\\\
y-10=x-12
\\\\
y=x-12+10
\\\\
y=x-2
.\end{array}
In the form $Ax+By=C$, the equation above is equivalent to
\begin{array}{l}\require{cancel}
y=x-2
\\\\
-x+y=-2
\\\\
x-y=2
.\end{array}
Hence, the different forms of the equation of the line with the given conditions are
\begin{array}{l}\require{cancel}
\text{slope-intercept form: }
y=x-2
\\\\
\text{standard form: }
x-y=2
.\end{array}