How do you tell whether the lines for each pair of equations are parallel, perpendicular, or neither: #y=-4x+3, -2x+8y=5#?

Question

Eva Abramson · Accepted Answer

#"lines are perpendicular"# #"the equation of a line in "color(blue)"slope-intercept form"# is. #•color(white)(x)y=mx+b# #"where m is the slope and b the y-intercept"# #"express both equations in this form and consider their"# #"slopes"# #• " parallel lines have equal slopes"# #• " the product of the slopes of perpendicular lines equals -1"# #y=-4x+3larrcolor(blue)"is in slope-intercept form"# #"with slope m "=-4# #"rearranging "-2x+8y=5larrcolor(blue)"add 2x to both sides"# #rArr8y=2x+5# #"divide all terms by 8"# #rArry=1/4x+5/8larrcolor(blue)"in slope-intercept form"# #"with slope "=1/4# #-4!=1/4" hence lines are not parallel"# #-4xx1/4=-1" hence lines are perpendicular"#

Genesis Allen · Answer

undefined To determine the relationship between the lines represented by each pair of equations, we compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. Otherwise, they are neither parallel nor perpendicular.

First equation: y = -4x + 3
Second equation: -2x + 8y = 5

To find the slope of the first equation, we compare it to the slope-intercept form (y = mx + b), where m is the slope. So, the slope of the first equation is -4.

To find the slope of the second equation, we rewrite it in slope-intercept form. First, isolate y: -2x + 8y = 5 → 8y = 2x + 5 → y = (2/8)x + (5/8). Therefore, the slope of the second equation is 2/8, which simplifies to 1/4.

Comparing the slopes:
- The slope of the first equation is -4.
- The slope of the second equation is 1/4.

Since the slopes are not equal and not negative reciprocals of each other, the lines represented by the equations are neither parallel nor perpendicular.