Answer
$x-y+6=0$
Work Step by Step
The equation of a line in slope-intercept form is:
$$y=mx+b,\tag1$$
where $m$ is the slope and $b$ in the $y$-intercept.
We are given:
- the line is parallel to the line passing through }P(2,5)\text{ and }Q(-2,1)
- $R(1,7)$ belongs to the line.
Because the line is parallel to the line passing through $P$ and $Q$, it means they have the same slope. We determine that slope:
$$\begin{align*}
m&=\dfrac{y_Q-y_P}{x_Q-x_P}\\
&=\dfrac{1-5}{-2-2}\\
&=1.
\end{align*}$$
We substitute $m=1$ in Eq. $(1)$:
$$y=x+b.\tag2$$
Determine $b$ using the coordinates of the point $R$ in Eq. $(2)$:
$$\begin{align*}
7&=1+b\\
b&=6.
\end{align*}$$
The equation of the line is:
$$y=x+6.$$
We write it in the general form $Ax+By+C=0$:
$$x-y+6=0.$$
