
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion 
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August 13th, 2017, 05:29 AM  #1 
Newbie Joined: Aug 2014 From: England Posts: 14 Thanks: 0 Math Focus: Calculus  System of equations (help)
Hi, I have a question I was hoping somebody could help with. I have two equations as follows: 2U + V = 7 UV =6 Now I could easily do this in my head and get the answer, but I want to solve it using the appropriate mathematical principles. If I turned the equation into a quadratic I could solve using a calculator and I get U = 1.5 and V = 4, or U =2 and V =3 However what if I did not use a calculator? How would I solve it then? What I did was rearrange the first equation to get V = 7  2U Plug that into the second equation to get U(7  2U) = 6 ... and expand to get 2U^2 + 7U  6 = 0 And I cannot go any further in solving for U without using a calculator 
August 13th, 2017, 05:32 AM  #2 
Newbie Joined: Aug 2014 From: England Posts: 14 Thanks: 0 Math Focus: Calculus 
@denis

August 13th, 2017, 05:41 AM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,397 Thanks: 479 Math Focus: Yet to find out. 
$2U^2 + 7U  6 = 0$ Factor U $U(7  2U)  6 = 0$ What values of U will cause the above to be true? Alternatively, you could use the quadratic formula. Have you learnt about that? Last edited by Joppy; August 13th, 2017 at 05:43 AM. 
August 13th, 2017, 05:41 AM  #4 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 872 Thanks: 60 Math Focus: सामान्य गणित 
Use quadratic formula $\displaystyle U = \frac {b\pm \sqrt {b^{2} 4ac}}{2a}$ For $\displaystyle aU^{2} +bU + c =0$ 
August 13th, 2017, 06:31 AM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,887 Thanks: 716  
August 13th, 2017, 07:52 AM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,397 Thanks: 479 Math Focus: Yet to find out.  
August 13th, 2017, 08:34 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,140 Thanks: 1415 
For distinct numbers p and q, the expression (x  p)(x  q) has two zeros: p and q. The expression can be written as x²  (p + q)x + pq. For the posted problem, p = 2U and q = V, so p + q = 7 and pq = 2UV = 12. Can one convert the expression x²  7x + 12 into the form (x  p)(x  q)? This would be a bit awkward if p and q can't be integers. If, however, they can be integers, their product is 12, and so there are only a very few possibilities to try. Those that work are (p, q) = (3, 4) and (p, q) = (4, 3), because the sum of p and q is then 7. As U = p/2 and V = q, the required solutions are (U, V) = (3/2, 4) and (U, V) = (2, 3). 

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