Answer
$\text{a) Slope-Intercept Form: }
y=\dfrac{2}{5}x-\dfrac{39}{5}
\\\text{b) Standard Form: }
2x-5y=39$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Find the slope of the line defined by the given equation, $
5x+2y=18
.$ Then use the Point-Slope Form of linear equations with the given point, $
(2,-7)
$ to find the equation of the needed line. Finally, Express the answer in both the Slope-Intercept and Standard forms.
$\bf{\text{Solution Details:}}$
In the form $y=mx+b,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
5x+2y=18
\\\\
2y=-5x+18
\\\\
\dfrac{2y}{2}=\dfrac{-5x}{2}+\dfrac{18}{2}
\\\\
y=-\dfrac{5}{2}x+9
.\end{array}
Using $y=mx+b$ or the Slope-Intercept Form, where $m$ is the slope, then the slope of the line with the equation above is
\begin{array}{l}\require{cancel}
m=-\dfrac{5}{2}
.\end{array}
Since parallel lines have negative reciprocal slopes, then the needed line has the following characteristics:
\begin{array}{l}\require{cancel}
m=\dfrac{2}{5}
\\\text{Through: }
(2,-7)
.\end{array}
Using $y-y_1=m(x-x_1)$ or the Point-Slope Form of linear equations, the equation of the line with the given conditions,
\begin{array}{l}\require{cancel}
y_1=
-7
,\\x_1=
2
,\\m=
\dfrac{2}{5}
,\end{array}
is
\begin{array}{l}\require{cancel}
y-y_1=m(x-x_1)
\\\\
y-(-7)=\dfrac{2}{5}(x-2)
\\\\
y+7=\dfrac{2}{5}(x-2)
.\end{array}
In the form $y=mx+b$ or the Slope-Intercept Form, the equation above is equivalent to
\begin{array}{l}\require{cancel}
y+7=\dfrac{2}{5}(x-2)
\\\\
y+7=\dfrac{2}{5}(x)+\dfrac{2}{5}(-2)
\\\\
y+7=\dfrac{2}{5}x-\dfrac{4}{5}
\\\\
y=\dfrac{2}{5}x-\dfrac{4}{5}-7
\\\\
y=\dfrac{2}{5}x-\dfrac{4}{5}-\dfrac{35}{5}
\\\\
y=\dfrac{2}{5}x-\dfrac{39}{5}
.\end{array}
In the form $ax+by=c$ or the Standard Form, the equation above is equivalent to
\begin{array}{l}\require{cancel}
5(y)=5\left( \dfrac{2}{5}x-\dfrac{39}{5} \right)
\\\\
5y=2x-39
\\\\
-2x+5y=-39
\\\\
-1(-2x+5y)=-1(-39)
\\\\
2x-5y=39
.\end{array}
Hence, $
\text{a) Slope-Intercept Form: }
y=\dfrac{2}{5}x-\dfrac{39}{5}
\\\text{b) Standard Form: }
2x-5y=39
.$
