Answer
(a) slope-intercept form
$\color{blue}{y=3x-7}$
(b) standard form
$\color{red}{3x-y=7}$
Work Step by Step
RECALL:
(1) The slope-intercept form of a line's equation is $y=mx+b$ where $m$=slope and $(0, b)$ is the line's y-intercept.
(2) The standard form of a line's equation is $Ax+By=C$ where $A\ge0$ and A, B, and C are integers.
(3) The point-slope form of a line's equation is $y-y_1=m(x-x_1)$ where $m$=slope and $(x_1, y_1)$ is a point on the line.
(4) The slope $m$ of the line that contains the points $(x_1. y_1)$ and $(x_2, y_2)$ is given by the formula:
$$m=\dfrac{y_2-y_1}{x_2-x_1}$$
(5) Parallel lines have the same slope.
The line is parallel to $3x-y=1$. The slope-intercept form of this equation is $y=3x-1$. Since the slope of this line is $3$, then the slope of the line parallel to is also $3$.
The line we are looking for has a slope of $3$ and contains the point $(2, -1)$.
Thus, the point-slope form of the line's equation is:
$$y-(-1)=3(x-2)
\\y+1=3(x-2)$$
(a) slope-intercept form
The slope-intercept form of the line can be derived from the equation above:
$y+1=3(x-2)
\\y+1=3(x)-3(2)
\\y+1=3x-6
\\y+1-1=3x-6-1
\\\color{blue}{y=3x-7}$
(b) standard form
The standard form of the line's equation can be derived from the slope-intercept form:
$y=3x-7
\\y+7=3x-7+7
\\y+7=3x
\\y+7-y=3x-y
\\7=3x-y
\\\color{red}{3x-y=7}$
