Determine if Parallel Are the lines defined by the equations y=-3x+1 and 2y=-6x-8 parallel?

Solution:

Step 1: Identify the Slopes

For the first equation, $y=-3x+1$, the slope is the coefficient of $x$, which is $-3$.

For the second equation, $2y=-6x-8$, we first divide both sides by 2 to get $y=-3x-4$. Here, the slope is also $-3$.

Step 2: Compare the Slopes

Since both lines have the same slope of $-3$, they are parallel.

Knowledge Notes:

To determine if two lines are parallel, one must compare their slopes. The slope of a line in the slope-intercept form, $y=mx+b$, is represented by the coefficient $m$. If two lines have the same slope, they are parallel, provided they are not the same line (which would make them coincident). In this case, the lines represented by the equations $y=-3x+1$ and $2y=-6x-8$ have the same slope of $-3$ after simplifying the second equation to slope-intercept form by dividing all terms by 2. Since their slopes match, the lines are parallel.