Two water pipelines are represented by the equations $kx + 3y +1 = 0$ and $2x + y + 3 = 0$. For what value of k, the pipelines cross each other?

Two lines cross each other when they intersect at exactly one point.

Lines DON'T cross when they are parallel (never meet) or coincident (same line, infinite points).

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Given lines:

Pipeline 1: $kx + 3y + 1 = 0$

Pipeline 2: $2x + y + 3 = 0$

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Two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are parallel when:

$\frac{a_1}{a_2} = \frac{b_1}{b_2}$

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For our pipelines:

$a_1 = k$, $b_1 = 3$, $c_1 = 1$

$a_2 = 2$, $b_2 = 1$, $c_2 = 3$

Lines are parallel when:

$\frac{k}{2} = \frac{3}{1}$

$k = 6$

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When $k = 6$, check if lines are coincident (same line).

Lines are coincident when:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Substituting $k = 6$:

$\frac{6}{2} = \frac{3}{1} = \frac{1}{3}$

$3 = 3 \neq \frac{1}{3}$

Since the ratios are not all equal, the lines are parallel but not coincident.

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When $k = 6$: Lines are parallel and do not cross.

When $k \neq 6$: Lines intersect at exactly one point.

Therefore, the pipelines cross each other for $k \neq 6$.