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To solve the problem, we need to find the equation of a line that passes through the point \((-4, -7)\) and is parallel to the line with equation \(y = 2x + 6\).

### Step 1: Understanding the slope of the given line
The equation of the line given is \(y = 2x + 6\), which is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

From the equation \(y = 2x + 6\), the slope \(m\) is 2.

### Step 2: Slope of the parallel line
Since the two lines are parallel, they will have the same slope. Therefore, the slope of the line passing through \((-4, -7)\) is also \(m = 2\).

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line. In this case, we know:
- The slope \(m = 2\)
- The point \((x_1, y_1) = (-4, -7)\)

Substitute these values into the point-slope form:
\[
y - (-7) = 2(x - (-4))
\]
Simplifying:
\[
y + 7 = 2(x + 4)
\]
Now distribute the 2 on the right-hand side:
\[
y + 7 = 2x + 8
\]
Subtract 7 from both sides:
\[
y = 2x + 1
\]

### Step 4: Compare with the options
The equation of the line is \(y = 2x + 1\), which corresponds to option **A**.

### Final Answer: A. \(y = 2x + 1\)

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:
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 would the equation change if the line was perpendicular instead of parallel?
4. What are the steps to convert a line equation from point-slope to slope-intercept form?
5. How do you verify that a point lies on a given line?

**Tip:** Always check that the slope of parallel lines is identical.

What is the equation of the line that passes through the point (-4, -7) and is parallel to the line with equation y = 2x + 6?

Find the equation of a line passing through the point (-4, -7) that is parallel to the line y = 2x + 6. This problem explores linear equations, the point-slope form, and the concept of parallel lines with equal slopes. Suitable for students in Grades 9-10.

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