**b.** What are 

x 1x1​, 

x 2x2​, 

and 

x 3x3 

when 

k 1k1equals=negative 11−11​, 

k 2k2equals=22​, 

and 

k 3k3equals=negative 81−81​?

x 1x1equals=enter your response here

x 2x2equals=enter your response here

x 3x3equals=enter your response here

For the given values \( k_1 = -11 \), \( k_2 = 2 \), and \( k_3 = -81 \), the solutions for \( x_1 \), \( x_2 \), and \( x_3 \) are:

\[
\begin{aligned}
x_1 & = -7 \\
x_2 & = -2 \\
x_3 & = 4
\end{aligned}
\]

Would you like further explanations or have any other questions? Here are some follow-up questions:

1. How would you interpret these values geometrically?
2. What happens to the solution if we change the sign of all constants \( k_1 \), \( k_2 \), and \( k_3 \)?
3. How does the consistency of the system affect the solution?
4. What does the solution tell you about the relationship between the variables?
5. How can you check if this solution satisfies the original equations?

**Tip:** When substituting new values into a matrix equation, it's important to recalculate the results carefully, as small changes in input can lead to different solutions.

Explore how to solve linear equations using the substitution method with a detailed example. This problem involves substituting given values for variables and solving for unknowns. Suitable for high school students in Grades 9-12.

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