SAT Question 14360f84: 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 distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form $Ax+By=C$, where $A$, $B$, and $C$ are constants, are parallel if the coefficients for $x$ and $y$ in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in the form $Ax+By=C$ by subtracting $9y$ from both sides of the equation to yield $4x-18y=5$. The second equation in the given system can be written in the form $Ax+By=C$ by subtracting $4x$ from both sides of the equation to yield $-4x+hy=2$. The coefficient of $x$ in this second equation, $-4$, is $-1$ times the coefficient of $x$ in the first equation, $4$. For the lines to be parallel, the coefficient of $y$ in the second equation, $h$, must also be $-1$ times the coefficient of $y$ in the first equation, $-18$. Thus, $h=-1(-18)$, or $h=18$. Therefore, if the given system has no solution, the value of $h$ is $18$.