Question

-x + 4y - 3z = -15
-4y - 3z = -11
x - 6y - z = -3

Explanation:

Step1: Add Eq1 and Eq3 to eliminate $x$

$(-x + 4y - 3z) + (x - 6y - z) = -15 + (-3)$
Simplify: $-2y - 4z = -18$, divide by -2: $y + 2z = 9$ (label as Eq4)

Step2: Rearrange Eq2 for $4y$

$-4y - 3z = -11 \implies 4y = -3z + 11$ (label as Eq5)

Step3: Solve Eq4 for $y$

$y = 9 - 2z$ (label as Eq6)

Step4: Substitute Eq6 into Eq5

$4(9 - 2z) = -3z + 11$
Expand: $36 - 8z = -3z + 11$

Step5: Solve for $z$

$36 - 11 = 8z - 3z \implies 25 = 5z \implies z = 5$

Step6: Substitute $z=5$ into Eq6

$y = 9 - 2(5) = 9 - 10 = -1$

Step7: Substitute $y=-1,z=5$ into Eq3

$x - 6(-1) - 5 = -3$
Simplify: $x + 6 - 5 = -3 \implies x + 1 = -3 \implies x = -4$

Answer:

$x=-4$, $y=-1$, $z=5$