Do the lines 3x + 4y + z = 1, 3x + 2y - 1 = 0 and x + y + z = 0 have a common point of intersection

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To determine if the three lines 3x + 4y + z = 1, 3x + 2y - 1 = 0 and x + y + z = 0 solve the three equations.

3x + 4y + z = 1 ...(1)

3x + 2y - 1 = 0 ...(2)

x + y + z = 0 ....(3)

From (2), x = (1 - 2y)/3

Substitute in (3)

=> (1 - 2y)/3 + y + z = 0

=> 1 - 2y + 3y + 3z = 0

=> z = (-1 - y)/3

Substitute in (1)

=> 3((1 - 2y)/3) + 4y + (-1 - y)/3 = 1

=> 1 - 2y + 4y + (-1 - y)/3 = 1

=> 2y + (-1 - y)/3 = 0

=> 6y - 1 - y = 0

=> y = 1/5

z = -2/5, x = 1/5

**The required point of intersection is (1/5, 1/5, -2/5)**

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