x + 2y + 3z = 2 2x + 3y - 42 = -5 3x + y + z = 3

To solve the system of equations, we will work with the following equations:

1. \( x + 2y + 3z = 2 \)
2. \( 2x + 3y - 42 = -5 \)
3. \( 3x + y + z = 3 \)

First, let's rewrite the second equation to isolate the variables:

\[ 2x + 3y = -5 + 42 \Rightarrow 2x + 3y = 37 \]

Now our system of equations is:

1. \( x + 2y + 3z = 2 \)
2. \( 2x + 3y = 37 \)
3. \( 3x + y + z = 3 \)

Let's express \( z \) in terms of \( x \) and \( y \) from the first equation (1):

\[ 3z = 2 - x - 2y \Rightarrow z = \frac{2 - x - 2y}{3} \]

Now substitute \( z \) into the third equation (3): \[ 3x + y + \frac{2 - x - 2y}{3} = 3 \]

To eliminate the fraction, multiply the entire equation by 3: \[ 9x + 3y + 2 - x - 2y = 9 \] Combine like terms: \[ 8x + y + 2 = 9 \Rightarrow 8x + y = 7 \tag{4} \]

Now, we have two equations to work with now. We can simplify our system to:

2. \( 2x + 3y = 37 \)
3. \( 8x + y = 7 \)

Now, solve for \( y \) from equation (4): \[ y = 7 - 8x \]

Substituting this expression for \( y \) in equation (2): \[ 2x + 3(7 - 8x) = 37 \] Simplify: \[ 2x + 21 - 24x = 37 \] Combine like terms: \[ -22x + 21 = 37 \] Subtract 21 from both sides: \[ -22x = 16 \Rightarrow x = -\frac{16}{22} = -\frac{8}{11} \]

Now substitute \( x \) back into the equation for \( y \): \[ y = 7 - 8\left(-\frac{8}{11}\right) = 7 + \frac{64}{11} \] Convert \( 7 \) to a fraction: \[ y = \frac{77}{11} + \frac{64}{11} = \frac{141}{11} \]

Finally, substitute \( x \) and \( y \) back into the equation for \( z \): \[ z = \frac{2 - \left(-\frac{8}{11}\right) - 2\left(\frac{141}{11}\right)}{3} \] Simplifying: \[ z = \frac{2 + \frac{8}{11} - \frac{282}{11}}{3} = \frac{2 - \frac{274}{11}}{3} = \frac{\frac{22}{11} - \frac{274}{11}}{3} = \frac{-\frac{252}{11}}{3} = -\frac{252}{33} = -\frac{84}{11} \]

Thus, the solution to the system of equations is: \[ x = -\frac{8}{11}, \quad y = \frac{141}{11}, \quad z = -\frac{84}{11} \]