Parallel, Perpendicular, or Neither: Neither Graph:
Practice makes perfect
Lines are parallel if their slopes are identical, and perpendicular if their slopes are negative reciprocals. Any other relationship between the lines would be neither parallel nor perpendicular.
Graphing the Lines
To get a rough idea as to whether the lines are parallel, perpendicular or neither, we might want to draw the lines first. In order to do that, we will find two points lying on each line. We can do this by substituting x=0 and x=1 into their formulas to obtain the y-value.
Line
x
y
Point
y=x+5
0
0+5= 5
( 0, 5)
y=x+5
1
1+5= 6
( 1, 6)
y=-5x-1
0
-5( 0)-1= -1
( 0, -1)
y=-5x-1
1
-5( 1)-1= -6
( -1, -6)
Now that we have two points per line, we can connect each pair to graph the given lines.
Looking at the graph, we can suspect that the lines are neither parallel nor perpendicular.
Determining the Relationship
In order to check whether the lines are perpendicular, let's write each equation in slope-intercept form, highlighting their slopes.
Given Equation
Slope-intercept form
Slope
y=x+5
y= 1x+5
m_1= 1
y=-5x-1
y= - 5x-1
m_2= -5
Since the lines have different slopes, we can conclude that they are not parallel. To determine whether or not they are perpendicular, we calculate the product of the slopes. Any two slopes whose product equals - 1 are negative reciprocals, and therefore the lines are perpendicular.
The slopes of the given lines are not negative reciprocals, so the lines are not perpendicular. Therefore, we can conclude that the lines are neither parallel nor perpendicular.
