To find the equation of a line that passes through a given point and is parallel to another line, we follow these steps:

Step 1: Identify the slope of the given line

The given equation of the line is $y = 2x + 6$. This is in slope-intercept form, $y = mx + b$, where $m$ is the slope.

Thus, the slope $m = 2$.

Step 2: Use the point-slope form

Since the lines are parallel, they have the same slope. Now, we'll use the point-slope form of the equation, which is:

$y - y_1 = m(x - x_1)$

where:

• $m = 2$ (slope of the parallel line),
• $(x_1, y_1) = (-4, -7)$ (the given point).

Substitute these values into the equation:

$y - (-7) = 2(x - (-4))$

This simplifies to:

$y + 7 = 2(x + 4)$

Step 3: Simplify the equation

Distribute the 2 on the right-hand side:

$y + 7 = 2x + 8$

Now, subtract 7 from both sides:

$y = 2x + 1$

Conclusion:

The equation of the line is $y = 2x + 1$, which matches option A.

Would you like further details or clarification?

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Here are 5 related questions:

1. How do you determine the slope of a line from its equation?
2. What is the significance of parallel lines having the same slope?
3. How do you convert a point-slope form equation to slope-intercept form?
4. Can two lines be parallel if their slopes are different?
5. How can you find the y-intercept of a line passing through a given point?

Tip: For lines that are perpendicular, their slopes are negative reciprocals of each other!