Q.Find the equation of the line passing through the point $(2, 3)$ and parallel to the line $4x + 5y = 6$.

A line parallel to $4x+5y=6$ keeps the same $x,y$ coefficients: $4x+5y=k$; find $k$ from the given point.

Any line parallel to $4x + 5y = 6$ has the same coefficients of $x$ and $y$, differing only in the constant term. So write the required line as

for some constant $k$ to be determined. Since it passes through $(2, 3)$, these coordinates must satisfy it:

Hence the line is $4x + 5y = 23$. …

Alternatively, find the slope of the given line as $-\tfrac{4}{5}$, then apply point-slope form: $y - 3 = -\tfrac{4}{5}(x - 2)$, which on clearing fractions …

Changing the coefficients of $x$ or $y$ (which would tilt the line and destroy parallelism), or using the constant $6$ from the original line in …