Which two numbers when added together yield $16$, and when multiplied together yield $55$.
I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I couldn't.
In $x+y=16$, I know $x=16/y$ but when I plug it back in I get something like $16/y + y = 16$, then I multiply the left side by $16$ to get $2y=256$ and then ultimately $y=128$. Am I doing something wrong?
Best Answer
Our two equations are: $$x + y = 16 \tag{1}$$ $$xy = 55\tag{2}$$
Rewriting equation (1) in terms of just $y =$ something, we get:
$$y = 16-x$$
Substituting this into equation (2) leaves us: $$x(16-x) = 55$$ $$16x-x^2=55 \implies x = 5 \ \ \text{or} \ \ 11$$
which can be easily seen by factoring or using the quadratic formula. It follows that $y=11|x=5$ and $y=5|x=11$.
Thus your solutions in terms of $(x,y)$ are $(5,11)$ and $(11,5)$.