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June 28th, 2018, 02:11 AM   #1
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Infinite solutions problem

I am trying to solve following problem. I am trying to find x,y,z in terms of $\displaystyle \lambda$. And then equating the resultant equation to 0. Am I correct?
Attached Images solutions.jpg (13.1 KB, 8 views)

Last edited by skipjack; June 28th, 2018 at 07:05 AM. June 28th, 2018, 07:46 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,124 Thanks: 2332 No, because "equating an equation to zero" doesn't make sense. What did you equate to zero? June 28th, 2018, 08:59 AM #3 Member   Joined: Aug 2017 From: India Posts: 54 Thanks: 2 After replacing y and z, finally I arrived at the following equation if I hopefully did not a mistake $\displaystyle x*\lambda*(\lambda^2+4*\lambda-3)=0$ So I find that there are 3 solutions, with one of them being 0. Am I correct? Please advise. Last edited by skipjack; June 28th, 2018 at 02:22 PM. June 28th, 2018, 02:25 PM #4 Global Moderator   Joined: May 2007 Posts: 6,856 Thanks: 745 Calculate the determinant: $\lambda^3+4\lambda-36=0$. This has only 1 real root, which is the desired solution. June 28th, 2018, 02:27 PM   #5
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Quote:
 Originally Posted by MathsLearner123 I hopefully did not a mistake . . .
Can you post all your work, so that the mistake(s) in it can be identified?

The three equations are satisfied by $x = y = z = 0$. In the right circumstances, there may be infinitely many other solutions. June 28th, 2018, 05:27 PM #6 Senior Member   Joined: Feb 2010 Posts: 714 Thanks: 151 These three equations represent three planes in 3 dimensional space. To have an infinite number of solutions these three planes must either be coincident or must intersect in a common line. The planes cannot be coincident since the attitude numbers (coefficients of x,y,z) are not proportional. To have a common line, the three planes must have two points in common. (0,0,0) is common to all three so all you need is one more. I think that $\displaystyle \left( \dfrac{-\lambda^2+4\lambda-8}{2},\lambda+2,8-\lambda \right)$ is also common to all three planes (although I'm not sure of my arithmetic). If I'm correct then it appears that any value of $\displaystyle \lambda$ would make a line with (0,0,0) that produces an infinite number of solutions. Can someone check this? June 28th, 2018, 08:49 PM #7 Global Moderator   Joined: Dec 2006 Posts: 21,124 Thanks: 2332 It's incorrect. June 29th, 2018, 08:50 AM   #8
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Quote:
 Originally Posted by MathsLearner123 After replacing y and z, finally I arrived at the following equation if I hopefully did not a mistake $\displaystyle x*\lambda*(\lambda^2+4*\lambda-3)=0$ So I find that there are 3 solutions, with one of them being 0. Am I correct? Please advise.
This is the correct conclusion to draw from your equation if the quadratic has two roots, yes. To satisfy the equation $x$, $\lambda$ or $(\lambda^2+4*\lambda-3)$ must be equal to zero. If neither $\lambda \ne 0$ and $(\lambda^2+4*\lambda-3)\ne 0$ you must have $x=0$ which leads to a single solution. On the other hand, if either $\lambda = 0$ or $(\lambda^2+4*\lambda-3) = 0$, the equation is satisfied by any value of $x$ and you thus have an infinite number of solutions.

Unfortunately, the quadratic has no real roots.

Last edited by v8archie; June 29th, 2018 at 08:52 AM. June 29th, 2018, 10:54 AM #9 Global Moderator   Joined: Dec 2006 Posts: 21,124 Thanks: 2332 The correct equation doesn't factorize that simply. June 29th, 2018, 12:01 PM   #10
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This is the attempt I have done. The final equation is
$\displaystyle x(\lambda^3+4\lambda-40)=0;$
I am not sure why I got a different equation last time. Sorry if the attachments are not clear. Next time, I will try to use math editor. Please advise whether I am correct.
Attached Images page2.jpg (8.5 KB, 3 views) Page1.jpg (70.2 KB, 6 views)

Last edited by skipjack; June 29th, 2018 at 10:05 PM. Tags infinite, problem, solutions Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post blimper Linear Algebra 4 August 30th, 2016 07:53 PM king.oslo Linear Algebra 4 April 6th, 2014 05:47 PM 84grandmarquis Linear Algebra 2 September 29th, 2013 02:59 PM EduXx Computer Science 2 October 7th, 2012 04:10 PM 84grandmarquis Algebra 0 December 31st, 1969 04:00 PM

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