Question

1.
Are the graphs of the lines in the pair parallel? Explain.

y = 6x + 9
27x – 3y = –81

Answer 1

To determine if the graphs of the given lines are parallel, we need to check their slopes. Two lines are parallel if they have the same slope.

1. The first line is already in slope-intercept form \(y = mx + b\):
\[
y = 6x + 9
\]
From this, we see that the slope \(m\) is 6.

2. Now, we will reformat the second line, given by the equation:
\[
27x - 3y = -81
\]
First, we can isolate \(y\) to put it into slope-intercept form. Start by moving \(27x\) to the other side:
\[
-3y = -27x - 81
\]
Next, divide through by -3 to solve for \(y\):
\[
y = 9x + 27
\]
From this equation, we see that the slope \(m\) of the second line is 9.

Now we can compare the slopes of both lines:
- Slope of the first line = 6
- Slope of the second line = 9

Since the slopes are different (6 ≠ 9), the lines are **not parallel**.