sat-dog

Choice D is correct. The x-coordinate $a$ of the x-intercept $\left(a,0\right)$ can be found by substituting $0$ for $y$ in the given equation, which gives $7x+2\left(0\right)=-31$, or $7x=-31$. Dividing both sides of this equation by $7$ yields $x=-\frac{31}{7}$. Therefore, the value of $a$ is $-\frac{31}{7}$. The y-coordinate $b$ of the y-intercept $\left(0,b\right)$ can be found by substituting $0$ for $x$ in the given equation, which gives $7\left(0\right)+2y=-31$, or $2y=-31$. Dividing both sides of this equation by $2$ yields $y=-\frac{31}{2}$. Therefore, the value of $b$ is $-\frac{31}{2}$. It follows that the value of $\frac{b}{a}$ is $\frac{-{\displaystyle \frac{31}{2}}}{-{\displaystyle \frac{31}{7}}}$, which is equivalent to $\left(\frac{31}{2}\right)\left(\frac{7}{31}\right)$, or $\frac{7}{2}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.