Answer
(a) slope-intercept form
$\\\color{blue}{y=\frac{5}{3}x+5}$
(b) standard form
$\color{red}{5x-3y=-15}$
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:
$$m=\dfrac{y_2-y_1}{x_2-x_1}$$
Solve for the slope using the formula in (4)above to obtain:
$$m=\dfrac{5-0}{0-(-3)}
\\m=\dfrac{5}{0+3}
\\m=\dfrac{5}{3}$$
(a) slope-intercept form
The line has a slope of $\dfrac{5}{3}$ and a y-intercept of $(0, 5)$.
Thus, the point-slope form of the line's equation is:
$$\\\color{blue}{y=\frac{5}{3}x+5}$$
(b) standard form
The standard form of the line's equation can be derived from the slope-intercept form:
$y=\frac{5}{3}x+5
\\3(y) = 3(\frac{5}{3}x+5)
\\3y=5x+15
\\3y-15=5x+15-15
\\3y-15=5x
\\3y-15-3y=5x-3y
\\-15=5x-3y
\\\color{red}{5x-3y=-15}$
