To determine which equation represents a line parallel to line $G$ passing through the point $(8, 5)$, we need to understand a few key concepts:

Parallel Lines

• Parallel lines have the same slope but different $y$-intercepts.
• Therefore, the equation of a line parallel to $G$ will have the same slope as $G$.

Since the specific slope of line $G$ is not given explicitly in the problem, we will infer it from the options.

Answer Choices Breakdown:

• Option A: $y = x + 8$ — This is the equation of a line with slope $1$ (because the coefficient of $x$ is 1). Its $y$-intercept is $8$.

• Option B: $y = 5$ — This is a horizontal line with a slope of 0, passing through $y = 5$ for all $x$.

• Option C: $y = x + 5$ — This is the equation of a line with slope $1$, and its $y$-intercept is $5$.

• Option D: $y = 8$ — This is another horizontal line with a slope of 0, passing through $y = 8$.

Conclusion:

The only two equations that could be parallel to line $G$ (assuming $G$ has a slope of 1) are A and C, because both have a slope of 1. However, since we want the line to pass through the point $(8, 5)$, let's check which one satisfies this condition:

• For $y = x + 5$, substitute $x = 8$: $y = 8 + 5 = 13$ So, the line does not pass through $(8, 5)$.

• For $y = x + 8$, substitute $x = 8$: $y = 8 + 8 = 16$ Again, this line does not pass through $(8, 5)$.

Thus, neither of these lines directly matches the point. Since this is unclear, the likely answer in general for parallel lines would have been A or C.

Please clarify if $G$'s slope might be known?

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Would you like further details or have any questions on this?

Here are five related questions to explore further:

1. What is the general form of a line equation and how is the slope calculated?
2. How do you determine the equation of a line given a point and a slope?
3. What distinguishes vertical lines from horizontal ones in terms of equations?
4. How do parallel and perpendicular lines relate in terms of their slopes?
5. How would the equation change if the slope of line $G$ was $-1$?

Tip: To find the equation of a line passing through a specific point, you can always use the point-slope form of a line equation: $y - y_1 = m(x - x_1)$.