Algebra

$$\frac{3x}{2} - \left(a - \frac{x}{2}\right) = 6$$ $$\frac{3x}{2} - a - \left(-\frac{x}{2}\right) = 6$$ $$\frac{3x}{2} - a + \frac{x}{2} = 6$$

Add $a$ to both sides.

$$\frac{3x}{2} + \frac{x}{2} = 6 + a$$ $$\frac{3x + x}{2} = 6 + a$$ $$\frac{4x}{2} = 6 + a$$ $$2x = 6 + a$$

Divide both sides by $2$.

$$x = \frac{6 + a}{2}$$

Let’s denote Claire’s points by $x$. According to the problem, Adam gets three more points than Claire. Hence, Adam gets $x + 3$ points. Similarly, Benjamin gets two times the difference between Claire’s points and $5$, which is equal to $2(x - 5)$ points. According to the problem, Adam and Benjamin get an equal number of points. Thus,

$$2(x - 5) = x + 3$$ $$2 \cdot x - 2 \cdot 5 = x + 3$$ $$2x - 10 = x + 3$$

Subtract $x$ from both sides.

$$2x - 10 - x = x + 3 - x$$ $$x - 10 = 3$$

Add $10$ to both sides.

$$x - 10 + 10 = 3 + 10$$ $$x = 13$$

Hence, Adam gets $x + 3 = 13 + 3 = 16$ points.

$$2x - 3(a - x) = 5x + 6$$ $$2x - 3 \cdot a - 3 \cdot (-x) = 5x + 6$$ $$2x - 3a + 3x = 5x + 6$$ $$5x - 3a = 5x + 6$$

Subtract $5x$ from both sides.

$$-3a = 6$$

Divide both sides by $-3$.

$$a = \frac{6}{-3}$$ $$a = -2$$

We can see that if $a = -2$, the equation is true for any value of $x$, and if $a \neq -2$, the equation is false for any value of $x$.

The number of small cups of coffee sold cannot be negative, and the same goes for the number of medium cups of coffee sold. This is represented by the first two inequalities

$$s \geq 0$$ $$m \geq 0$$

which are the same in each of the options.

According to the problem, the cafe sells more than $200$ cups of coffee. This is represented by the inequality

$$s + m > 200$$

The inequality $s + m \geq 200$ implies that the number of cups of coffee sold could also be equal to $200$, which is not correct according to the problem.

According to the problem, the sales exceed $660$ dollars. Each small cup costs $2.80$ dollars, so the total sales from small cups of coffee is $2.80s$. Similarly, the sales from medium cups of coffee is $3.60m$. Hence, the inequality

$$2.80s + 3.60m > 660$$

correctly represents this situation.

Graphing the three inequalities, the point in the solution region where the integer values of $x$ and $y$ are greatest is $(2, -1)$. Hence, the maximum value of $x + y$ is $2 + (-1) = 1$.