Geometry

The equation of the reflection of the line y = 2x + 1 across the line y = 5x - 3.

Intersection:

$f_1(x)=f_2(x)\\ 2x+1=5x-3\\ 3x=4\\ x_{1,2}=\frac{4}{3}\\ y_{1,2}=\frac{11}{3}=3\frac{2}{3}\\ P_{1,2}(1\frac{1}{3},3\frac{2}{3})$

gradient

$m_{1,2}=tan\ [\frac{1}{2}(\arctan 2+\arctan 5)]\\ m_{1,2}=2.9145$

Point direction equation

$y=m(x-x_P)+y_P \\ y=2.914536(x-1.\overline {33})+3.\overline{66}\\ y=2.914536x-3.886053+3.666667$

The equation of the reflection is

$\color{blue}f_{1,2}(x)= y=2.914536x-0.219381$

 !