line segments

[bdyoung8]

Ok, so it has been 15 years since my last math class and I have lost a lot of my geometry knowledge. I know that the slope of a line is y=mx+b, but I have forgotten how to express a line segment. So if I wanted to plot out a line segment from (-7,2) to (-6,-4), what would that look like?

 

[HallsofIvy]

Ok, so it has been 15 years since my last math class and I have lost a lot of my geometry knowledge. I know that the slope of a line is y=mx+b

Well, strictly speaking, that is the "slope-intercept" form of the equation. The number, m, is the "slope".

, but I have forgotten how to express a line segment. So if I wanted to plot out a line segment from (-7,2) to (-6,-4), what would that look like?

"Express" a line segment? Do you mean something like "y= mx+ b \(\displaystyle -7\le x\le -6\)? Of course, if y= 2 when x= -7, "y= mx+ b" becomes 2= -7m+ b and if y= -4 when x= - 6, -4= -6m+ b. We can eliminate b by subtracting one equation from the other: 2- (-4)= 6= -7m- (-6m)= -m so m= -6. Then 2= -7m+ b becomes 2= 42+ b so that be b= -40.

The line through (-7, 2) and (-6, -4) has equation y= -6x- 40. The line segment between those two points would be expressed as "y= -6x- 40 for \(\displaystyle -7\le x\le -6\).

As far as plotting it is concerned, draw an x-y coordinate system, two perpendicular axes with x= 0, y= 0 at the intersection, x values increasing to the right and decreasing to the left, y values increasing upward and decreasing downward. Mark the points (-7, 2) and (-6, -4). Go left from the origin, where the two axes cross, 7 places (x= -7) and up two places (y= 2) and make a dot marking the point (-7, 2). Similarly go left from the origin 6 places (x= -6) and down 4 places (y= -4) and make a dot marking the (-6, -4). Finally, use some kind of straight edge to draw the straight line between those two dots.

 

[pka]

Ok, so it has been 15 years since my last math class and I have lost a lot of my geometry knowledge. I know that the slope of a line is y=mx+b, but I have forgotten how to express a line segment. So if I wanted to plot out a line segment from (-7,2) to (-6,-4), what would that look like?

The subject line does say "line segment".
Here is that line segment: \(\displaystyle \{(-7+t,2-6t) :0\le t\le 1\}\)

 

[bdyoung8]

The subject line does say "line segment".
Here is that line segment: \(\displaystyle \{(-7+t,2-6t) :0\le t\le 1\}\)

Thank you very much, would you be willing to explain how you came about this so that I can learn to how to do this myself?

 

[pka]

Thank you very much, would you be willing to explain how you came about this so that I can learn to how to do this myself?

It really is a simple idea that comes from vector geometry. If \(\displaystyle P~\&~ Q\) are two points then \(\displaystyle P+t(Q-P)\) is the line segment between the two points. You see for \(\displaystyle t=0\) you get \(\displaystyle P\) for \(\displaystyle t=1\) you get \(\displaystyle Q\). It is really useful, for \(\displaystyle t=0.5\) you get the mid-point of the segment.

Here for \(\displaystyle P: (-7,2),~Q: (-6,-4)~\&~Q-P=(1,-6)\) so for \(\displaystyle (-7,-4)+t(1,-6)=(1+t,2-6t)\).