Question

The equation $y-k=m\left( x-h \right)$ in which only $m$ and $h$ are fixed represents what?

Answer 1

Hint: Break the terms in the right hand side and also take $k$ to the right hand side. Pair up this $k$ with $mh$ and observe the equation carefully, the equation will represent a family of parallel straight lines.

“Complete step-by-step answer:”
In geometry, parallel lines are lines which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. In other words, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel. A line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. The parallel symbol is $\parallel $. For example: $AB\parallel CD$ indicates that line AB is parallel to CD.
Now, we come to the question. We have been provided with the equation $y-k=m\left( x-h \right)$. This can be written as:
$\begin{align}
  & y-k=m\left( x-h \right) \\
 & y-k=mx-mh \\
 & y=mx-mh+k \\
 & y=mx+\left( k-mh \right) \\
\end{align}$
Now, considering $m$ as the slope and $\left( k-mh \right)$ as the intercept, the given equation represents a straight line with slope $m$. Now, it is given that $m\text{ and }h$ are constant, that means the slope of the given line is constant. We can see that, as $k$ varies the intercept of the line varies. So, there are lines with equal slope and different intercepts. These lines therefore, represent a family of parallel straight lines.

Note: We have considered the family of lines parallel because they are having the same value of $m$ as it is fixed. Only the intercept will vary due to $k$, that means different lines will cut the $y-axis$ at different places.