Consider the system of linear equations: $$\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + k z &= m \end{aligned}$$ If the system has an infinite number of solutions, what is the value of $(k + m)$?

Answer Key:

13

### Solution: Subtracting row 2 from row 3: $(k - 3)z = m - 10$. For an infinite number of solutions, row 3 must become a linear combination of rows 1 and 2, requiring $k - 3 = 0 \implies k = 3$ and $m - 10 = 0 \implies m = 10$. Thus $k + m = 3 + 10 = 13$. Correct: 13.

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