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Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion


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August 13th, 2017, 05:29 AM   #1
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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
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August 13th, 2017, 05:32 AM   #2
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@denis
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August 13th, 2017, 05:41 AM   #3
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$-2U^2 + 7U - 6 = 0$

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.
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August 13th, 2017, 05:41 AM   #4
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Use quadratic formula
$\displaystyle U = \frac {-b\pm \sqrt {b^{2} -4ac}}{2a}$

For
$\displaystyle aU^{2} +bU + c =0$
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August 13th, 2017, 06:31 AM   #5
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Originally Posted by srahman33 View Post
@denis
WHAT did I do now?
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August 13th, 2017, 07:52 AM   #6
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Originally Posted by Denis View Post
WHAT did I do now?
Tsk tsk!
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August 13th, 2017, 08:34 AM   #7
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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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