Why is the point-slope form useful if you want to write the equation of a line through a point that is parallel to a given line.

Recall what a parallel line is; two lines that are parallel are defined as having the same gradient or slope. Consider a line:

y = mx + b

If we want to find a certain line that is / parallel / to the original line passing through an arbitrary point (x₁, y₁), it is useful to understand the point-gradient or point-slope formula.

The gradient to the line y = mx + b is simply m. So, any parallel line to y = mx + b will have the same gradient. Examples include: y = mx + 1, y = mx + 200, y = mx + g

All we need to know, now, is to identify what specific line hits the desired point. Well, the point-gradient formula can help with that. Recall that the point-gradient formula is:

y - y₀ = m(x - x₀), where (x₀, y₀) is the point of interest.
Hence, it is useful to use the point-slope formula when asked for a point and a set of parallel lines to the original line.