Clever System of Equations Problem

It might not be obvious, but dummy variables will make this problem much easier. Let $a=x+y$ and $b=\frac{x}{y}$ . Then: $a+b=19$ $a \cdot b=60$ This looks much easier to solve! Guess and check will be enough to get the answer. We’re looking for two numbers that add up to $19$ and multiply to $60$ , just like factoring a quadratic. We find that $15$ and $4$ work for these equations, and since addition and multiplication are both commutative, we have two solutions that both include the same integers: $(15,4)$ and $(4,15)$ . Now we have two cases:

Case 1: $a=15, b=4$ $x+y=15$ $\frac{x}{y}=4$ $x=4y$ $5y=15$ $\boxed{x=12, y=3}$ Case 2: $a=4, b=15$ $x+y=4$ $\frac{x}{y}=15$ $x=15y$ $16y=4$ $\boxed{x=\frac{15}{4}, y=\frac14}$ Therefore, from our two solutions, $(12,3)$ and $(\frac{15}{4}, \frac14)$ , $\boxed{\textbf{a+b+c+d+x+y=39}}$

Let $p = x+y$ and $q = \frac{x}{y}$ . We have that $yq = x$ , implying that $y = \frac{p}{1+q}$ and $x = \frac{pq}{1+q}$ . Now, since $x+y+\frac{y}{x} = A$ and $\frac{x}{y}(x+y) = B$ , we have $p+q = A$ anbd $pq = B$ .

Now, consider $P(z) = (z - p)(z - q) = z^2 - (p+q)z + pq = z^2-Az+B$ . This polynomial has roots $p$ and $q$ that can be found by the quadtratic formula. For $A = 19$ and $B = 60$ , we get the following cases:

• $p = \frac{A+\sqrt{A^2-4B}}{2} = \frac{19+11}{2} = 15,\hspace{4mm} q = \frac{A-\sqrt{A^2-4B}}{2} = 4$ , implying that $(x,y) = (12, 3)$

  • $p = 4, \hspace{4mm} q = 15$ , implying that $(x,y) = (\frac{15}{4}, \frac{1}{4})$

So we input $12+3+4+4+1+15 = 39$ .

x + y + x/y = 19 (1) x(x+y)/y = 60 (2) In (2): x + y = 60y/x (3) Sub. (3) into (1): 60y/x + x/y = 19 60y^2 - 19xy + x^2 = 0 (x - 15y)(x - 4y) = 0 x = 15y or x = 4y Case 1, x = 15y then in (2): 15y(16y)/y = 60 240y = 60 y = 1/4 x = 15(1/4) = 15/4 So, a = 15, b = 4, c = 1 and d = 4 Case 2, x = 4y then in (2): 4y(5y)/y = 60 20y = 60 y = 3 x = 4(3) = 12 So, x = 12 and y = 3 Therefore, x + y + a + b + c + d = 39

x+y = 4, x/y = 15, so y=1/4 , x=15/4 or x+y = 15, x/y = 4, y=3, x=12
So a+b+c+d+x+y = 39