Answer
Point-slope form: $y+9=7(x-5)$
Slope-intercept form: $ y=7x-44$
or $f(x)=7x-44$.
Work Step by Step
In order to determine the equation of a line in point-slope form we need the slope $m$ and a point $(x_1,y_1)$ on it:
$y-y_1=m(x-x_1)$...........(1)
We are given the point $(5,-9)$, so we have to find the slope $m$.
We are given the equation of a perpendicular line on our line:
$\Rightarrow x+7y=12$
Let $m_1$ be the slope of this line.
Two lines are perpendicular if their slopes are negative reciprocal to each other, therefore $m$ and $m_1$ check the equation:
$m\cdot m_1=-1$......... (2).
Determine the slope $m_1$. First bring this line's equation to the slope-intercept form.
Subtract $x$ from both sides.
$\Rightarrow x+7y-x=12-x$
Simplify.
$\Rightarrow 7y=12-x$
Divide both sides by $7$.
$\Rightarrow \frac{7y}{7}=\frac{12-x}{7}$
Simplify.
$\Rightarrow y=\frac{12}{7}-\frac{x}{7}$
Rearrange.
$\Rightarrow y=-\frac{1}{7}x+\frac{12}{7}$
This is in the standard form of slope-intercept form
$y=m_1x+c$, where $m_1$ is a slope of the line.
We have $m_1=-\frac{1}{7}$.
Substitute $m_1=-\frac{1}{7}$ in equation (2) to determine the slope $m$:
$m\cdot \left(-\frac{1}{7}\right)=-1$
Multiply by $-7$:
$m=7$
Since the line passes through a point $(x_1,y_1)$ then the point-slope form of our line's equation is determined by plugging $(x_1,y_1)=(5,-9)$ and $m=7$ in equation (1):
$\Rightarrow y-(-9)=(7)(x-5)$
Simplify.
$\Rightarrow y+9=7(x-5)$
The above equation is the point-slope form.
Now subtract $9$ from both sides.
$\Rightarrow y+9-9=7(x-5)-9$
Simplify.
$\Rightarrow y=7x-35-9$
$\Rightarrow y=7x-44$
The above equation is the slope-intercept form.
or we can write $f(x)=7x-44$.
