To determine the value of $k$ that makes the given line parallel to the line

$3x - 5y + 47 = 0$

we need to find the slope of this given line.

Step 1: Find the slope of the given line

Rearrange it into slope-intercept form $y = mx + b$:

$-5y = -3x - 47$

$y = \frac{3}{5}x + \frac{47}{5}$

So, the slope is $\frac{3}{5}$.

Step 2: Find $k$ in equation $6x + ky = 18$

Rearrange into slope-intercept form:

$ky = -6x + 18$

$y = \frac{-6}{k}x + \frac{18}{k}$

For the lines to be parallel, the slopes must be equal:

$\frac{-6}{k} = \frac{3}{5}$

Step 3: Solve for $k$

Cross multiply:

$-6 \times 5 = 3 \times k$

$-30 = 3k$

$k = -10$

Thus, the required value of $k$ is $\mathbf{-10}$.

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Would you like further clarification or more details?

Related Questions:

1. What is the condition for two lines to be parallel?
2. How do you find the slope of a line from its equation?
3. How do you convert an equation into slope-intercept form?
4. What happens if two lines have the same slope but different y-intercepts?
5. Can two parallel lines ever intersect? Why or why not?

Tip: When comparing slopes, always convert equations into the form $y = mx + b$ to identify the slope easily.