To find the equation of a line that passes through the point $(18, 2)$ and is parallel to the line $3y - x = -12$, we need to:

Step 1: Find the Slope of the Given Line

Convert the equation $3y - x = -12$ into slope-intercept form $y = mx + b$ to determine the slope.

1. Start with the original equation: $3y - x = -12$
2. Add $x$ to both sides: $3y = x - 12$
3. Divide by 3: $y = \frac{1}{3}x - 4$

Now, we see that the slope $m$ of this line is $\frac{1}{3}$.

Step 2: Use the Point-Slope Form for the Parallel Line

Since parallel lines have the same slope, the slope of our desired line is also $\frac{1}{3}$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (18, 2)$ and $m = \frac{1}{3}$, we get:

$y - 2 = \frac{1}{3}(x - 18)$

Step 3: Simplify the Equation

Expand and simplify: $y - 2 = \frac{1}{3}x - 6$ $y = \frac{1}{3}x - 4$

Final Answer

The equation of the line that passes through $(18, 2)$ and is parallel to $3y - x = -12$ is: $y = \frac{1}{3}x - 4$

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Related Questions

1. How would you find the equation of a line perpendicular to $3y - x = -12$ that passes through $(18, 2)$?
2. How can you determine if two lines are parallel without converting to slope-intercept form?
3. What would change in the equation if the point given was $(18, -2)$ instead of $(18, 2)$?
4. How would you find the equation of a line with a given slope that passes through two specific points?
5. How do you derive the general form of a line's equation from the point-slope form?

Tip

Remember, for parallel lines, the slopes are identical. For perpendicular lines, the slopes are negative reciprocals of each other.