SAT Question e142f83c: Systems of Linear Equations answer and explanation with Desmos solution

Choice B is correct. It's given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Adding $6$ to both sides of $y-6=\frac{4}{5}\left(x+5\right)$ yields $y=\frac{4}{5}\left(x+5\right)+6$. Applying the distributive property to this equation yields $y=\frac{4}{5}x+\frac{20}{5}+6$, or $y=\frac{4}{5}x+10$. Adding $v$ to both sides of $y-v=\frac{4}{5}\left(x+10\right)$ yields $y=\frac{4}{5}\left(x+10\right)+v$. Applying the distributive property to this equation yields $y=\frac{4}{5}x+\frac{40}{5}+v$, or $y=\frac{4}{5}x+8+v$. For the equations $y=\frac{4}{5}x+10$ and $y=\frac{4}{5}x+8+v$ to be equivalent, it follows that $10=8+v$. Subtracting $8$ from both sides of this equation yields $2=v$. Therefore, the value of $v$ is $2$.