Determine which of the following lines, if any, are parallel or perpendicular:

line a: 3y+x=6 line b: y=3x+2
line c: 3x+y=0

parallel lines have the same slope

perpendicular have slopes that are opposite reciprocals

change each line equation to the form y = mx + b and you will be able to tell.

To determine if lines are parallel or perpendicular, we need to compare their slopes.

The slope-intercept form of a line is y = mx + b, where m is the slope of the line.

Let's determine the slopes of the given lines:

Line a: 3y + x = 6
Rewriting this equation in slope-intercept form: 3y = -x + 6
Dividing both sides by 3: y = (-1/3)x + 2/3
The slope of line a is -1/3.

Line b: y = 3x + 2
The slope of line b is 3.

Line c: 3x + y = 0
Rewriting this equation in slope-intercept form: y = -3x
The slope of line c is -3.

Now, let's determine the relationships between the lines:

Line a is parallel to line b if their slopes are equal.
-1/3 ≠ 3, so line a is not parallel to line b.

Line a is perpendicular to line b if the product of their slopes is -1.
(-1/3)(3) = -1, so line a is perpendicular to line b.

Line a is parallel to line c if their slopes are equal.
-1/3 ≠ -3, so line a is not parallel to line c.

Line a is perpendicular to line c if the product of their slopes is -1.
(-1/3)(-3) = 1, so line a is perpendicular to line c.

In summary:
- Line a is perpendicular to line b.
- Line a is perpendicular to line c.

To determine whether two lines are parallel or perpendicular, we need to compare their slopes.

The slope-intercept form of a line is y = mx + b, where m is the slope of the line.

For line a: 3y + x = 6, we can rearrange the equation to be in slope-intercept form:
3y = -x + 6
y = (-1/3)x + 2/3

The slope of line a is -1/3.

For line b: y = 3x + 2, the given equation is already in slope-intercept form. The slope of line b is 3.

For line c: 3x + y = 0, we can rearrange the equation to be in slope-intercept form:
y = -3x

The slope of line c is -3.

Now, let's compare the slopes of these lines:

The slope of line a is -1/3.
The slope of line b is 3.
The slope of line c is -3.

Since none of the slopes are equal, none of the given lines are parallel.

To determine if any of the lines are perpendicular, we need to check if the product of their slopes is -1.

The product of the slopes of line a and line b is (-1/3)(3) = -1, which means line a and line b are perpendicular.

The product of the slopes of line b and line c is (3)(-3) = -9, which means line b and line c are not perpendicular.

Therefore, only line a and line b are perpendicular.