Solution:
\({\text{y}} = 12{\text{x}} - 8\)

Solution:
\({\text{y}} = 2{\text{x}} + 2\)
Details:
We will use the point-slope form to find the equation of the line. Point-slope form is the following:

\({\text{y}}{-}{\text{y}}_1={\text{m}}({\text{x}}-{\text{x}}_1)\)

We will use the point \(({\color{Red}1}, {\color{Purple}4})\) and the slope of \({\color{DarkOrange}2}\) and substitute them in to \({\text{y}}{-}{\color{Purple}{\text{y}}_1}={\color{DarkOrange}{\text{m}}}({\text{x}}-{\color{Red}{\text{x}}_1})\):

\({\text{y}}{-}{\color{Purple}4}={\color{DarkOrange}2}({\text{x}}-{\color{Red}1})\)

Then distribute on the right side:

\({\text{y}}{-}{\color{Purple}4}={\color{DarkOrange}2}{\text{x}}-{\color{DarkOrange}2}(1)\)

Which simplifies to the following:

\({\text{y}}{-}{4}=2{\text{x}}{-}{2}\)

Add 4 to both sides:

\({\text{y}}{-}{4}{\color{Red}+4} = 2{\text{x}} - 2 {\color{Red}+4}\)

Which gives us the equation:

\(y = 2x +{\color{Red}2}\)

Solution:
\({\text{y}} =-\dfrac{2}{9}{\text{x}}+10\)

Solution:
\({\text{y}}=-\dfrac{5}{11}{\text{x}}-3\)

Solution:
\((0,4)\)
Details:
We will use the point-slope form to find the equation of the line. Point-slope form is the following:

\({\text{y}}{-}{\text{y}_{1}}=\text{m}\left ( {\text{x}}{-}{\text{x}_{1}} \right )\)

We will use the point \(({\color{Red}10}, {\color{Purple}6})\) and the slope of \({\color{DarkOrange} \dfrac{1}{5}}\) and substitute them in to the following:

\({\text{y}}{-}{\color{Purple} {\text{y}_{1}}}={\color{DarkOrange} \text{m}}\left ( {\text{x}}{-}{\color{Red} {\text{x}_{1}}} \right )\)

Then distribute on the right side:

\({\text{y}}{-}{\color{Purple}6}={\color{Orange}\dfrac{1}{5}}{\text{x}}-{\color{Orange}\dfrac{1}{5}}(10)\)

Which simplifies to the following:

\({\text{y}}{-}6={\color{Orange}\dfrac{1}{5}}{\text{x}}-2\)

Add 6 to both sides:

\({\text{y}}{-}6{\color{Red}+6}={\color{Orange}\dfrac{1}{5}}{\text{x}}-2{\color{Red}+6}\)

Which gives us the equation:

\(\text{y}={\color{DarkOrange} \dfrac{1}{5}}\text{x}+{\color{Red} 4}\)

The equation is in slope-intercept form, \(y=mx+b\), so we can see that the y-intercept is at \(4\) or the point \((0,4)\).

Solution:
\((0,11)\)