• Perpendicular lines ,have negative reciprocal slopes, meaning that if line 1 is perpendicular to line 2, then the slope of line 2 is:
[tex]m_2=-\frac{1}{m_1}[/tex]• Parallel lines ,have the same slope, meaning that is line 1 is parallel to line e, then the slope of line 2 is:
[tex]m_2=m_1[/tex]Procedure
To be able to compare each straight-line equation, we have to homogenize the form in which they are written. For example, choosing the slope-intercept form:
[tex]y=mx+b[/tex]where m is the slope and b is the y-intercept.
6.
In this case, the first straight-line equation is written in the slope-intercept form:
[tex]y=-2x+1[/tex]where m1 = -2.
However, we have to isolate y from the second equation in order to have it in the slope-intercept form:
[tex]2x-4y=4[/tex][tex]-4y=-2x+4[/tex][tex]y=\frac{-2x+4}{-4}[/tex][tex]y=\frac{1}{2}x-1[/tex]where m2 = 1/2.
If we compare these slopes:
[tex]m_2=-\frac{1}{m_1}=-\frac{1}{-2}=\frac{1}{2}[/tex]we can see that these lines are perpendicular.
8.
In this case, neither of the lines are in the slope-intercept. Thus, we have to convert them by isolating y:
• First equation
[tex]y=-x+3[/tex]• Second equation
[tex]y=x-5[/tex]Again, comparing the slopes:
[tex]m_2=-\frac{1}{m_1}=-\frac{1}{-1}=1[/tex]therefore, these will show perpendicular lines.
10.
The first equation is in the slope-intercept form, but we have to change the second one:
[tex]y=\frac{3}{2}x-1[/tex]In this case, if we compare the slopes:
[tex]m_2=-\frac{1}{m_1}=-\frac{1}{\frac{2}{3}}=-\frac{3}{2}[/tex]as this is not the case, these are not perpendicular lines. Also:
[tex]m_2\ne m_1[/tex]thus, these are not parallel lines. Then, these are neither perpendicular nor parallel.
Answer:
• 6. Perpendicular
,• 8. Perpendicular
,• 10. Neither
