2022 CEMC Cayley Problems/Problem 15

Problem

The integers $a$ , $b$ , and $c$ satisfy the equations $a + 5 = b$ and $5 + b = c$ and $b + c = a$ . The value of $b$ is

$\text{ (A) }\ -30 \qquad\text{ (B) }\ -20 \qquad\text{ (C) }\ -10 \qquad\text{ (D) }\ 0 \qquad\text{ (E) }\ 5$

Plugging the first equation into the last, we have:

$a + 5 + c = a$

Subtracting $a$ and $5$ , we see:

$c = -5$

We can then plug this into the second equation:

$5 + b = -5$

Subtracting $5$ from both sides, we see:

$b = \boxed {\textbf {(C) } -10}$

~anabel.disher

We can add all of the equations together. This gives:

$a + 5 + 5 + b + b + c = b + c + a$

Subtracting $a$ , $b$ , and $c$ , we get:

$5 + 5 + b = 0$

Simplifying the left side and moving it to the right, we have:

$b = \boxed {\textbf {(C) } -10}$

~anabel.disher