To determine the equation of the line passing through $(0, 1)$ and parallel to the line $x + 2y = 6$, let's follow these steps:

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Step 1: Determine the slope of the given line

The given line is: $x + 2y = 6$ Rewriting it in slope-intercept form $y = mx + c$: $2y = -x + 6 \implies y = -\frac{1}{2}x + 3$ The slope ($m$) of the given line is: $m = -\frac{1}{2}$

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Step 2: Identify the slope of the parallel line

Lines that are parallel have the same slope. Thus, the slope of the required line is also: $m = -\frac{1}{2}$

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Step 3: Use the point-slope form

The line passes through the point $(0, 1)$. Using the point-slope form of a line equation: $y - y_1 = m(x - x_1)$ Substitute $m = -\frac{1}{2}$, $(x_1, y_1) = (0, 1)$: $y - 1 = -\frac{1}{2}(x - 0)$ Simplify: $y - 1 = -\frac{1}{2}x \implies y = -\frac{1}{2}x + 1$

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Final Answer:

The equation of the required line is: $\boxed{y = -\frac{1}{2}x + 1}$

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From the options provided:

The correct answer is: $\boxed{y = -\frac{1}{2}x + 1}$

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Would you like a detailed explanation of the point-slope method or any other clarification?

Related Questions:

1. How do you find the slope of a line from a general equation like $Ax + By + C = 0$?
2. What is the difference between parallel and perpendicular slopes?
3. How do you convert between point-slope form and slope-intercept form of a line?
4. Can you derive the equation of a line perpendicular to $x + 2y = 6$ passing through a given point?
5. What are some applications of parallel lines in geometry or physics?

Tip:

For parallel lines, always remember that the slopes are equal. Adjust only the y-intercept based on the specific point the line passes through!