Solving A Nice System of Equations
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Substitute z=2^x and y=5-x into the first equation and rearrange to z^2-12z+32=0.
Thus, z=2^x=(12±4)/2=8 or 4, x=3 or 2, and y=2 or 3.
This shows that these are the only two real solutions.
Beautiful explanation, Sir...Thank You...
You are most welcome
I solved it with the first method. Not much of a challenge for most regular viewers, but that's because you do a great job building our abilities.
Do two lines equal by y:
1.y=log2(12-2^x)
2.y=5-x
=> 5-x=(first thing)
Both are downing => max two solutions for x => 2,3
I used the third method, and then the first method to make sure 2 and 3 are the only solutions.
The same. I then watched to see what the second method was. Turned out to be the first method wearing a false beard and moustache.
x = 3, y = 2 and vice versa.
Can't it be done by log?? I am unable to solve using log can anyone help???
i would define f(x) as f(x) = 2^x + 2^(5-x) and notice that f is decreasing for x < 2.5 and increasing for x > 2.5. That should suffice to show that x = 2,3 are the only solutions for f(x) = 12.
Syber seems to like churning through a lot of algebra.
But wait, Rick, you need Calculus to figure out when f is increasing!
My response would be
a) it seems weirdly constraining to allow algebra but not Calculus
b) you really don't need Calculus here. (But I would use it anyway.)
You can tell from the thumbnail without doing any math, I know 4 + 8 = 12 so maybe x is 2 or 3 while y is the other one and they of course add to be 5.
but how can you prove that it's the only solution
@@technolan6015 Well even if you used his method on other problems, it wouldn't prove they are the only 2 solutions.
It is obviously 2 and 3. The only problem here is to show it is the only solution (2,3 and 3,2 - they are symmetrical). As a teacher I do not appreciate the method demonstrated. The students should SEE the solution immediarely
He doesn't care about answers, it's about the method and new ways to solve. As a teacher you should also know it.
@@asheredude There is value in seeing the obvious answers. That's the practical way of how math is done in the real world. And, sure, it's important to show that your set of answers is complete. But it's also important to avoid going down rabbit holes.
@@rickdesper So didn't he do that in 3rd method, I mean it's all about the new ways, isn't it? There are many questions, it's not necessary to get the answer by just looking at it in each question. I agree with you even before starting the video, I also had 2 and 3 in my mind. But the comment of @ckrylova said," he do not appreciate these methods"
@@asheredude Dear commenter, the method is ok, but not in a trivial case. The method should be adequate to the problem not to shoot the flies with the missile.
As a teacher I see my mission in teaching students thinking, not applying methods as robots.
@@ckrylova But who says to apply the methods, he is just informing we have these methods to solve, and we will obviously apply the method which is short. If he would just do it like that then, the video will be of 1 minute. And talking about thinking, this was a case of obvious answer what about something like this question and you just have to solve them. But you said the method is 'ok' then What am I writing for, Have a good day !
I guess there's a lot of wisdom in the methods that you used but to me perhaps if you would use larger numbers a more analytical approach would be appropriate. Since 12 is such a small number one only has to look at the various powers of two and only two four or eight are less than 12 so it's immediately obvious that 8 + 4 = 12 and 2 + 3 = 5. Had you use much larger numbers where this was not so obvious then the approaches you took might be more appropriate. It took me 15 seconds to solve it before I even press play on the video