Math Snap

STEP 1

1. We have a system of three linear equations with three variables: $x$, $y$, and $z$.
2. We need to determine if each ordered triple is a solution to the system by substituting the values into each equation.
3. An ordered triple is a solution if it satisfies all three equations.

STEP 2

1. Substitute the ordered triple (1, 1, -1) into each equation and verify.
2. Substitute the ordered triple (3, 1, -2) into each equation and verify.

STEP 3

Substitute $(x, y, z) = (1, 1, -1)$ into the first equation $x + y + 2z = 0$:
$$1 + 1 + 2(-1) = 0$$ Simplify:
$$1 + 1 - 2 = 0$$ $$0 = 0$$ The first equation is satisfied.

STEP 4

Substitute $(x, y, z) = (1, 1, -1)$ into the second equation $-5x - 3y + 2z = -10$:
$$-5(1) - 3(1) + 2(-1) = -10$$ Simplify:
$$-5 - 3 - 2 = -10$$ $$-10 = -10$$ The second equation is satisfied.

STEP 5

Substitute $(x, y, z) = (1, 1, -1)$ into the third equation $5x - 2y + 3z = 0$:
$$5(1) - 2(1) + 3(-1) = 0$$ Simplify:
$$5 - 2 - 3 = 0$$ $$0 = 0$$ The third equation is satisfied.
Since all three equations are satisfied, the ordered triple $(1, 1, -1)$ is a solution.

STEP 6

Substitute $(x, y, z) = (3, 1, -2)$ into the first equation $x + y + 2z = 0$:
$$3 + 1 + 2(-2) = 0$$ Simplify:
$$3 + 1 - 4 = 0$$ $$0 = 0$$ The first equation is satisfied.

Substitute $(x, y, z) = (3, 1, -2)$ into the second equation $-5x - 3y + 2z = -10$:
$$-5(3) - 3(1) + 2(-2) = -10$$ Simplify:
$$-15 - 3 - 4 = -10$$ $$-22 \neq -10$$ The second equation is not satisfied.
Since the second equation is not satisfied, the ordered triple $(3, 1, -2)$ is not a solution.
The ordered triple $(1, 1, -1)$ is a solution, while $(3, 1, -2)$ is not a solution.