SAT Question 59813abf: Systems of Linear Equations answer and explanation with Desmos solution

Choice D is correct. A system of two linear equations in two variables, $x$ and $y$, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, $Ax+By=C$ and $Dx+Ey=F$, are parallel and distinct if the coefficients of $x$ and $y$ in one equation are proportional to the corresponding coefficients in the other equation, meaning $\frac{D}{A}=\frac{E}{B}$, and the constants are not proportional, meaning $\frac{F}{C}$ is not equal to $\frac{D}{A}$ nor $\frac{E}{B}$. It's given that one of the equations in a system of two linear equations is $38(x-n)=38y+38n$, where $n$ is a positive constant. Distributing $38$ on the left-hand side of this equation yields $38x-38n=38y+38n$. Adding $38n$ to both sides of this equation yields $38x=38y+76n$. Subtracting $38y$ from both sides of this equation yields $38x-38y=76n$. Therefore, the given equation can be written in the form $Ax+By=C$, where $A=38$, $B=-38$, and $C=76n$. The equation in choice D, $4x-4y=4n$, is written in the form $Dx+Ey=F$, where $D=4$, $E=-4$, and $F=4n$. Using these values, it follows that the value of $\frac{D}{A}$ is $\frac{4}{38}$, or $\frac{2}{19}$; the value of $\frac{E}{B}$ is $\frac{-4}{-38}$, or $\frac{2}{19}$; and the value of $\frac{F}{C}$ is $\frac{4n}{76n}$, or $\frac{1}{19}$, since $n$ is a positive constant. Since $\frac{D}{A}=\frac{E}{B}$ and since $\frac{F}{C}$ is not equal to $\frac{D}{A}$ nor $\frac{E}{B}$, it follows that the lines represented by these two equations are parallel and distinct. Therefore, the second equation in this system of equations with no solution could be $4x-4y=4n$.