To determine if $(-1, 1, -2)$ satisfies each equation in the system, we’ll substitute $x_1 = -1$, $x_2 = 1$, and $x_3 = -2$ into each equation and check if they hold true.

The system of equations is:

1. $5x_1 + 2x_2 + x_3 = -5$
2. $6x_1 + 3x_2 + x_3 = -5$
3. $4x_1 + 2x_2 + x_3 = -3$

Step-by-Step Substitution

Equation 1

$5(-1) + 2(1) + (-2) = -5$ Calculating each term: $-5 + 2 - 2 = -5$ The first equation is satisfied.

Equation 2

$6(-1) + 3(1) + (-2) = -5$ Calculating each term: $-6 + 3 - 2 = -5$ The second equation is also satisfied.

Equation 3

$4(-1) + 2(1) + (-2) = -3$ Calculating each term: $-4 + 2 - 2 = -4$ The third equation is not satisfied, as $-4 \neq -3$.

Conclusion

Since $(-1, 1, -2)$ does not satisfy all three equations (specifically the third equation), it is not a solution to the system.

Would you like a more detailed breakdown or have any questions on this solution?

Here are some related questions for further practice:

1. How do you determine if a point satisfies a system of equations?
2. What methods can solve a system of linear equations?
3. Can a point partially satisfy a system of equations?
4. What is the geometric interpretation of a solution to a system of equations?
5. How can you verify a solution using matrix operations?

Tip: When substituting values into equations, double-check each arithmetic operation to ensure accuracy.