7 8 y - 5 8 x = 4 7 - 7 8 y 5 4

The correct answer is $\frac{7}{2}$. 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. Two lines represented by equations in standard 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, $\frac{7}{8}y-\frac{5}{8}x=\frac{4}{7}-\frac{7}{8}y$, can be written in standard form by adding $\frac{7}{8}y$ to both sides of the equation, which yields $\frac{14}{8}y-\frac{5}{8}x=\frac{4}{7}$, or $-\frac{5}{8}x+\frac{14}{8}y=\frac{4}{7}$. Multiplying each term in this equation by $-8$ yields $5x-14y=-\frac{32}{7}$. The second equation in the given system, $\frac{5}{4}x+\frac{7}{4}=py+\frac{15}{4}$, can be written in standard form by subtracting $\frac{7}{4}$ and $py$ from both sides of the equation, which yields $\frac{5}{4}x-py=\frac{8}{4}$. Multiplying each term in this equation by $4$ yields $5x-4py=8$. The coefficient of $x$ in the first equation, $5x-14y=-\frac{32}{7}$, is equal to the coefficient of $x$ in the second equation, $5x-4py=8$. For the lines to be parallel, and for the coefficients for $x$ and $y$ in one equation to be proportional to the corresponding coefficients in the other equation, the coefficient of $y$ in the second equation must also be equal to the coefficient of $y$ in the first equation. Therefore, $-14=-4p$. Dividing both sides of this equation by $-4$ yields $\frac{-14}{-4}=p$, or $p=\frac{7}{2}$. Therefore, if the given system of equations has no solution, the value of $p$ is $\frac{7}{2}$. Note that 7/2 and 3.5 are examples of ways to enter a correct answer.