For the system of linear equations

$x + y + z = 5000$
$6x + 7y+ 8z = 35800$
$6x + 7y-8z= 7000$

the values of $x, y$ and $z$ are:

$x=1000; y=2200; z=1800$

The correct answer is Option (3) → $x=1000; y=2200; z=1800$

6x + 7y + 8z = 35800

6x + 7y - 8z = 7000

Add the two equations: 12x + 14y = 42800 → 6x + 7y = 21400

Subtract the two equations: 16z = 28800 → z = 1800

Use x + y + z = 5000 → x + y = 3200

6x + 7y = 21400, x + y = 3200 → x = 3200 - y

6(3200 - y) + 7y = 21400 → 19200 - 6y + 7y = 21400 → y = 2200

x = 3200 - 2200 = 1000

x = 1000, y = 2200, z = 1800