We are asked to solve the system of linear equations: $x + y = a$ $y + z = b$ $z + x = c$

1. Problem Description

We are asked to solve the system of linear equations:

$x + y = a$

$y + z = b$

$z + x = c$

2. Solution Steps

We have the following system of equations:

$x + y = a$ (1)

$y + z = b$ (2)

$z + x = c$ (3)

Adding equations (1), (2) and (3), we get:

$x + y + y + z + z + x = a + b + c$

$2x + 2y + 2z = a + b + c$

$2(x + y + z) = a + b + c$

$x + y + z = \frac{a + b + c}{2}$ (4)

From equation (4), we can find $z$:

$z = (x + y + z) - (x + y)$

$z = \frac{a + b + c}{2} - a$

$z = \frac{a + b + c - 2a}{2}$

$z = \frac{-a + b + c}{2}$

From equation (4), we can find $x$:

$x = (x + y + z) - (y + z)$

$x = \frac{a + b + c}{2} - b$

$x = \frac{a + b + c - 2b}{2}$

$x = \frac{a - b + c}{2}$

From equation (4), we can find $y$:

$y = (x + y + z) - (z + x)$

$y = \frac{a + b + c}{2} - c$

$y = \frac{a + b + c - 2c}{2}$

$y = \frac{a + b - c}{2}$

3. Final Answer

$x = \frac{a - b + c}{2}$

$y = \frac{a + b - c}{2}$

$z = \frac{-a + b + c}{2}$