Find general solution for the system of linear equations

Question

2x+4y+z=3

x+2y+2z=3

x+2y+z=2

Answer 1

Find general solution for the system of linear equations:

2x + 4y + z = 3 ==> (1)

x + 2y + 2z = 3 ==> (2)

x + 2y + z = 2 ==> (3)

(2) - (3) ==> (x + 2y + 2z = 3) - (x + 2y + z = 2) ==> (2z = 3) - (z = 2) ==> z = 1

z = 1, substitute into (1) ==> 2x + 4y + z = 3 ==> 2x + 4y + 1 = 3 ==> 2x + 4y = 2 ==> x + 2y = 1 ==> x = 1 - 2y

General Solution: (x, y, z) = (1 - 2y, y, 1)

Check for example for y = 5, x = 1 - 2y ==> x = 1 - 2(5) ==> x = 1 - 10 ==> x = -9, z = 1

2x + 4y + z = 3 ==> (1): 2(-9) + 4(5) + 1 = 3 ==> -18 + 20 + 1 = 3 ==> 3 = 3

x + 2y + 2z = 3 ==> (2): -9 + 2(5) + 2(1) = 3 ==> -9 + 10 + 2 = 3 ==> 3 = 3

x + 2y + z = 2 ==> (3): -9 + 2(5) + 1 = 2 ==> -9 + 10 + 1 = 2 ==> 2 = 2

Check for example for y = 100, x = 1 - 2y ==> x = 1 - 2(100) ==> x = 1 - 200 ==> x = -199, z = 1

2x + 4y + z = 3 ==> (1): 2(-199) + 4(100) + 1 = 3 ==> -398 + 400 + 1 = 3 ==> 3 = 3

x + 2y + 2z = 3 ==> (2): -199 + 2(100) + 2(1) = 3 ==> -199 + 200 + 2 = 3 ==> 3 = 3

x + 2y + z = 2 ==> (3): -199 + 2(100) + 1 = 2 ==> -199 + 200 + 1 = 2 ==> 2 = 2

Note: It should work for any value of y. You can check any value of y such as: y = 0, y = -73, y = -2,435, y = -10,729, y = 783, y = 1,251,392, etc.

Answer 2

Subtract the last equation from the second to get z=1. Substitute z=1 into the first and second equations and notice they are not independent differing only by a constant factor of 2 ... Thus an infinite number of solutions are possible.