Answer
(-1, 1, -1, 1)
Work Step by Step
The question asks to find the solution to the system of equations.
Given:
1. $x + y + z + w = 0$
2. $x + y + 2z + 2w = 0$
3. $2x + 2y + 3z + 4w = 1$
4. $2x + 3y + 4z + 5w = 2$
Subtract equation 1 from 2, equation 3 from 4
$x + y + 2z + 2w = 0$
-($x + y + z + w = 0$)
$z + w = 0$
$2x + 3y + 4z + 5w = 2$
-($2x + 2y + 3z + 4w = 1$)
$y + z + w = 1$
Since $z + w = 0$, then $y = 1$
Substitute y = 1 to all equations
1. $x + 1 + z + w = 0$
2. $x + 1 + 2z + 2w = 0$
3. $2x + 2 + 3z + 4w = 1$
4. $2x + 3 + 4z + 5w = 2$
1. $x + z + w = -1$
2. $x + 2z + 2w = -1$
3. $2x + 3z + 4w = -1$
4. $2x + 4z + 5w = -1$
Multiply equation 2 by 2, then subtract it from equation 4
$2x + 4z + 5w = -1$
-($2x + 4z + 4w = -2$)
$w = 1$
Since $z + w = 0$, then $z = -1$
Solve for x
1. $x + 1 + -1 + 1 = 0$
$x = -1$
Solution: (-1, 1, -1, 1)
