If the system of linear equations \( x + y + z = 1 \), \( 2x + 2y + 3z = 6 \), \( x + 4y + 9z = 3 \) has a unique solution \( x = \alpha, y = \beta, z = \gamma \), then the value of \( \beta \) is:
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To save time, check if simple elimination is faster. From (1) \( x+y+z=1 \) and (2) \( 2x+2y+3z=6 \), we can see \( 2(x+y)+3z=6 \Rightarrow 2(1-z)+3z=6 \Rightarrow z=4 \). Substituting \( z=4 \) back into the equations makes solving for \( y \) much simpler.