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We need to write the equation of a line that is parallel to the given line \( 6y = -12x - 21 \) and passes through the point \((-4, 9)\).

### Step 1: Identify the slope of the given line
The given equation is in standard form. First, we convert it to slope-intercept form \( y = mx + b \), where \( m \) is the slope.

Starting with:
\[
6y = -12x - 21
\]

Divide through by 6 to isolate \( y \):
\[
y = -2x - \frac{21}{6}
\]
\[
y = -2x - 3.5
\]
So, the slope \( m \) of the given line is \( -2 \).

### Step 2: Use the slope for the parallel line
Parallel lines have the same slope, so the slope of the line we are looking for is also \( m = -2 \).

### Step 3: Find the equation of the new line
Now we use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m = -2 \), and \( (x_1, y_1) = (-4, 9) \).

Substitute the values:
\[
y - 9 = -2(x + 4)
\]
\[
y - 9 = -2x - 8
\]
\[
y = -2x + 1
\]

### Step 4: Match the equation with the answer choices
The equation of the line is \( y = -2x + 1 \), but this does not match any of the given choices exactly. Let's check the closest form among the choices. None of them are exactly parallel to the slope \( m = -2 \), as seen in the form \( y = mx + b \).

It seems the correct form should be given but it does not match any of the answer options directly. Based on the calculation, the answer should have the slope \( -2 \), which could indicate an issue with the answer options.

Would you like further clarification on any part of this solution?

### 5 Related Questions:
1. What is the slope of a line that is perpendicular to a given line with slope \( -2 \)?
2. How do you find the equation of a line perpendicular to \( 6y = -12x - 21 \)?
3. How can we check if two lines are parallel using their equations?
4. How do we convert a line equation from standard form to slope-intercept form?
5. What is the difference between the point-slope form and slope-intercept form of a line?

### Tip:
Always remember that parallel lines have identical slopes. When writing equations for parallel lines, simply ensure the slope is the same as the given line!

Write the equation of the line that is parallel to the given line described by the equation 6y = -12x - 21 and passes through the point (-4, 9).

Learn how to find the equation of a line that is parallel to a given line described by the equation 6y = -12x - 21 and passes through the point (-4, 9). This problem explores key algebraic concepts such as linear equations, slope-intercept form, and the parallel line theorem, suitable for students in Grades 9-10.

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