Perpendicular lines have slopes that are negative reciprocals of each other.

A perpendicular line has a negative reciprocal slope to the given line.

The given line, $\displaystyle y=\frac{-1}{2}x+4$, has a slope of $\displaystyle -\frac{1}{2}$ , as $\displaystyle m$ is the slope in the standard form equation  $\displaystyle (y=mx+b)$.

Slope of perpendicular line: $\displaystyle m=2$

Point: $\displaystyle (0,6)$

Using the point slope formula, we can solve for the equation:

 $\displaystyle y-y_{1}=m(x-x_{1})$

$\displaystyle y-6=2(x-0)$

$\displaystyle y-6=2x$

$\displaystyle y=2x+6$

Given

$\displaystyle \small h(x)=-\frac{5}3x-12$

We need a perpendicular line going through (14,0). 

Perpendicular lines have opposite reciprocal slopes.

So we get our slope to be

 $\displaystyle \small \frac{3}{5}$

Next, plug in all our knowns into $\displaystyle y=mx+b$ and solve for $\displaystyle b$.

$\displaystyle 0=14\cdot \frac{3}{5}+b$

$\displaystyle \small b=-\frac{42}{5}$.

Making our answer

$\displaystyle \small y=\frac{3x}{5}-\frac{42}{5}$.

Given a line $\displaystyle a$ defined by the equation $\displaystyle f(x)=mx+b$ with slope $\displaystyle m$, any line that is perpendicular to $\displaystyle a$ must have a slope$\displaystyle -\frac{1}{m}$, or the negative reciprocal of $\displaystyle m$.

Since $\displaystyle f(x)=4x+9$, the slope $\displaystyle m$ is $\displaystyle 4$ and the slope of any line $\displaystyle g(x)$ parallel to $\displaystyle f(x)$ must have a slope of $\displaystyle -\frac{1}{m}=-\frac{1}{4}$.

Since $\displaystyle g(x)$ also needs to have a $\displaystyle y$-intercept of $\displaystyle 7$, then the equation for $\displaystyle g(x)$ must be $\displaystyle g(x)=-\frac{1}{4}x+7$.

Given a line $\displaystyle a$ defined by the equation $\displaystyle f(x)=mx+b$ with slope $\displaystyle m$, any line that is perpendicular to $\displaystyle a$ must have a slope$\displaystyle -\frac{1}{m}$, or the negative reciprocal of $\displaystyle m$.

Since $\displaystyle f(x)=-5x-4$, the slope $\displaystyle m$ is $\displaystyle -5$ and the slope of any line $\displaystyle g(x)$ parallel to $\displaystyle f(x)$ must have a slope of $\displaystyle -\frac{1}{m}=-\frac{1}{-5}=\frac{1}{5}$.

Since $\displaystyle g(x)$ also needs to have a $\displaystyle y$-intercept of $\displaystyle 8$, then the equation for $\displaystyle g(x)$ must be $\displaystyle g(x)=\frac{1}{5}x+8$.

Given a line $\displaystyle a$ defined by the equation $\displaystyle f(x)=mx+b$ with slope $\displaystyle m$, any line that is perpendicular to $\displaystyle a$ must have a slope$\displaystyle -\frac{1}{m}$, or the negative reciprocal of $\displaystyle m$.

Since $\displaystyle f(x)=7x+2$, the slope $\displaystyle m$ is $\displaystyle 7$ and the slope of any line $\displaystyle g(x)$ parallel to $\displaystyle f(x)$ must have a slope of $\displaystyle -\frac{1}{m}=-\frac{1}{7}$.

Since $\displaystyle g(x)$ also needs to have a $\displaystyle y$-intercept of $\displaystyle -9$, then the equation for $\displaystyle g(x)$ must be $\displaystyle g(x)=-\frac{1}{7}x-9$.

The equation of a line in standard form is written as follows:

$\displaystyle y=mx+b$

Where $\displaystyle m$ is the slope of the line and $\displaystyle b$ is the y intercept. First, we can determine the slope of the perpendicular line using the knowledge that its slope must be the negative reciprocal of the slope of the line to which it is perpendicular. For the given line, we can see that $\displaystyle m=3$, so the slope of a line perpendicular to it will be the negative reciprocal of that value, which gives us:

$\displaystyle m_{perp}=-\frac{1}{m}=-\frac{1}{3}$

Now that we know the slope of the perpendicular line, we can plug its value into the formula for a line along with the coordinates of the given point, allowing us to calculate the $\displaystyle y$-intercept, $\displaystyle b$:

$\displaystyle 4=-\frac{1}{3}(2)+b$

$\displaystyle b=\frac{14}{3}$

We now have the slope and the $\displaystyle y$-intercept of the perpendicular line, which is all we need to write its equation in standard form:

$\displaystyle y=-\frac{1}{3}x+\frac{14}{3}$