Solution:
\({\text{y}}=-4{\text{x}}-14\)

Solution:
\({\text{y}}=\dfrac{1}{3}{\text{x}}+7\)

Solution:
\({\text{y}}=\dfrac{1}{8}{\text{x}}-\dfrac{3}{8}\)

Solution:
\({\text{y}}=9{\text{x}}-36\)
Details:
To change the equation into slope-intercept form, we write it in the form \(y=mx+b\).

Start with the original equation:

\({\text{x}}=\dfrac{{\text{y}}+36}{9}\)

We want to isolate the y, so our first step is to multiply both sides by 9.

\(9{\text{x}}=\left(\dfrac{{\text{y}}+36}{9}\right)9\)

Then cancel out the 9’s on the right side.

\(9{\text{x}}=\left(\dfrac{{\text{y}}+36}{\:\cancel{9}}\right)\:\cancel{9}\)

Which gives us:

\({\text{9x=y+36}}\)

Then subtract 36 from both sides.

\(9{\text{x}} {\color{Red}{-}36} = {\text{y}} + 36 {\color{Red}{-}36}\)

Which simplifies to:

\(9{\text{x}}{-}36={\text{y}}\)

Which is the same equation as:

\({\text{y}}=9{\text{x}}-36\)

Solution:
\({\text{y}}=-3{\text{x}}+\dfrac{7}{2}\)
Details:
To change the equation into slope-intercept form, we need to write it in the form

\(y=mx+b\)

Start with our original equation:

\(-6{\text{x}}{-}{2}\text{y}=-7\)

We want to isolate the term with \(y\), so we add \(6x\) to both sides:

\(-6\text{x}+6{\text{x}}{-}{2}\text{y}=-7+6\text{x}\)

Then simplify to:

\(-2\text{y}=-7+6\text{x}\)

Now we can multiply both sides by \({\color{Red} -\dfrac{1}{2}}\) (or divide by \(−2\) which is an equivalent operation).

\({\color{Red} -\dfrac{1}{2}}\left ( -2\text{y} \right )= {\color{Red} -\dfrac{1}{2}}\left ( -7+6\text{x} \right )\)

Multiply on the left and distribute on the right to get:

\({\color{Red}1}\text{y}= {\color{Red} -\dfrac{1}{2}}\left (-7 \right )+{\color{Red} -\dfrac{1}{2}}\left (6\text{x} \right )\)

Which gives us:

\({\text{y}}=\dfrac{7}{2}-\dfrac{6}{2}{\text{x}}\)

Which simplifies to:

\({\text{y}}=\dfrac{7}{2}-{\color{Red} 3}{\text{x}}\)

Then we can rewrite the equation so that it is in slope-intercept form:

\({\text{y}}=-3{\text{x}}+\dfrac{7}{2}\)

Solution:
\({\text{y}}=\dfrac{3}{2}{\text{x}}+\dfrac{1}{2}\)