Answer
Answer $C$: $(7,7)$
Work Step by Step
1) determine the equations of the line with slope $-6$ and passing through the point $(9,-5)$
2) check if the points belong to the line.
$\textbf{1) Determine the equation of the line}$
The equation of the line with slope $m$ and passing through the point $(x_0,y_0)$ is:
$$y-y_0=m(x-x_0).$$
Substitute $m=-6$ and $(x_0,y_0)=(9,-5)$ in the above equation:
$$\begin{align*}
y-(-5)&=-6(x-9)\\
y+5&=-6x+54\\
y&=-6x+54-5\\
y&=-6x+49.
\end{align*}$$
$\textbf{2) Check which of the points belong to the line}$
We take each point and substitute the coordinates of the point in the equation of the line.
A) $(6,10)$
$$\begin{align*}
10&\stackrel{?}{=}-6(6)+49\\
10&\stackrel{?}{=}13\\
10&\not=13.
\end{align*}$$
B) $(6,6)$
$$\begin{align*}
6&\stackrel{?}{=}-6(6)+49\\
6&\stackrel{?}{=}13\\
6&\not=13.
\end{align*}$$
C) $(7,7)$
$$\begin{align*}
7&\stackrel{?}{=}-6(7)+49\\
7&\stackrel{?}{=}-42+49\\
10&=7\checkmark.
\end{align*}$$
D) $(6,-4)$
$$\begin{align*}
-4&\stackrel{?}{=}-6(6)+49\\
-4&\stackrel{?}{=}13\\
-4&\not=13.
\end{align*}$$
So the point which fits is $(7,7)$. The correct answer is Answer $C$.
