My dad called this evening with a math problem from Jessica that they couldn't work out. Of course, he didn't have the problem in front of him at the time, so there wasn't much I could do, but they emailed it to me. I promised to give them a solution, along with the amount of time it took to figure it out. The two emails follow:
20)
x+y+z=4
5x+5y+5z=12
x-4y+z=9
My response:
3 seconds.
The problem here is that you have two equations that aren't consistant with each other. Those being the first and second equations. Think of it this way:
take the second equation
5x+5y+5z=12
and factor five out of the right side
5(x+y+z)=12
However, we know that x+y+z=4 from the first equation. Substitute that in and you get:
5(4)=12
Which is clearly not true. This is an inconsistant set of equations that has no solution. If you think about systems of equations visually then two equations with 2 unknowns are like looking for an intersection of 2 lines in the Cartesian plane. Most lines intersect, but it is possible for lines to be parallel. In 3 dimensions it suddenly becomes very easy to imagine 3 lines that do not share a single point. In terms of matricies (which Jessica should be introduced to this year and be tortured into using) the matrix describing this system would be singular, or have a determinant of zero, or be non-invertible, or have a non-zero null space or . . . the list goes on, and we've drifted into the realm of math 343 at BYU, which no one really understands.
So there you have it, in case you didn't already know.
6 comments:
Wow. See, I would have spent 20 minutes trying to solve it. (Because I do derive some pleasure in solving equations, pretty much because it's so rare.) Then, being totally unsuccessfull, I would assume it was my fault and give up.
Which I did.
I just recently spent an evening with one of my tutoring "students" doing problems like these. I think it would have taken me a bit longer than you to figure this one out. I was ready to start adding and subtracting equations to isolate variables. Were they trying to figure out if it was a real system or not or were they actually supposed to be solving it and maybe it was a typo? Boy, are we nerdy! Lavish praise!
Congrats, you've taken linear algebra at BYU. For those of you interested in such things, the determinate of the system is 0, which is just another way of saying that the system is inconsistent. BTW you misspelled consistent. Still lavish praise is deserved.
Ah, if only that were the only thing I spelled wrong.
Um, spellcheck? Don't you read my blog?
yeah yeah. the one stinkin' time I forget to spell check and everyone's all over me about it.
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