Determine if Parallel Are the lines defined by the equations y=-3x+1 and 2y=-6x-8 parallel?
The problem requires an examination of the slopes of the two lines represented by the equations y = -3x + 1 and 2y = -6x - 8. The question asks whether these two lines are parallel to each other. To determine if lines are parallel, their slopes must be equal, and the y-intercepts do not matter for this determination. Lines are parallel if they never intersect, which geometrically means they have the same slope when graphed in the coordinate plane.
Are the lines defined by the equations$y = - 3 x + 1$and$2 y = - 6 x - 8$parallel?
Answer
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.
