How do I prove that the gradient between a fixed and any general point on a given line is $m$?

Question

I encountered a confusing question in an A-level textbook that I'm currently studying from, it is as follows:

Let $P$ with coordinates $(p, q)$, be a fixed point on the 'curve' with equation $y = mx + c$ and let $Q$, with coordinates $(r, s)$, be any other point on $y = mx + c$. Use the fact that the coordinates of $P$ and $Q$ satisfy the equation $y = mx + c$ to show that the gradient of $PQ$ is $m$ for all positions of $Q$.

I do not know how to approach this, so any suggestions would be helpful.

Answer 1

Since $(p,q)$ is on the "curve" defined by the equation $y=m+c$, then substituting $x=p$ and $y=q$ we get

$$q = mp + c$$

Since $(r,s)$ is on the same curve, we also get

$$s=mr+c$$

Therefore we may rewrite the points as $P(p,mp+c)$ and $ Q(r,mr+c)$.

Now use the definition of slope to get an expression for the slope of the line segment between the points $P(p,mp+c)$ and $ Q(r,mr+c)$. Simplify that expression and the final formula for the slope will be very simple and finish the problem. I'll leave that final step to you.