48 x - 64 y = 48 y + 24 r y = 1 8 -

The correct answer is $-28$. 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 for $x$ and $y$ in the other equation. The first equation in the given system, $48x-64y=48y+24$, can be written in the form $Ax+By=C$ by subtracting $48y$ from both sides of the equation to yield $48x-112y=24$. The second equation in the given system, $ry=\frac{1}{8}-12x$, can be written in the form $Ax+By=C$ by adding $12x$ to both sides of the equation to yield $12x+ry=\frac{1}{8}$. The coefficient of $x$ in the second equation is $\frac{1}{4}$ times the coefficient of $x$ in the first equation. That is, $48\left(\frac{1}{4}\right)=12$. For the lines to be parallel, the coefficient of $y$ in the second equation must also be $\frac{1}{4}$ times the coefficient of $y$ in the first equation. Therefore, $-112\left(\frac{1}{4}\right)=r$, or $-28=r$. Thus, if the given system has no solution, the value of $r$ is $-28$.