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Old 07-30-2007   #1
Miyu
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Default What is the process in Algebra if there are two unknown numbers?

Example: The sum of two numbers is 45. The sum of their quotient & its reciprocal is 2.05. What is the product of these two numbers?
 
Old 07-30-2007   #2
cllau74
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x+y = 45 ...(1)x/y+y/x= 2.05(x^2+y^2)/xy = 2.05[(x+y)^2-2xy] /xy =2.05(2025 - 2xy) = 2.05xy4.05xy = 2025 xy = 500 ...(2)Solve (1) and (2)x+500/x = 45x^2-45x+500=0(x-25)(x-20) = 0x= 25 or 20y = 20 or 25.
 
Old 07-30-2007   #3
kael
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hi there...^,^if there are two unknowns then most probably you'll have 2 equations to solve that problem.so here goes...lets take the first no. as x and then the second one as y...for simplicitythe first eq isx + y = 45 (eq1)the second eq would bex/y + y/x = 2.05 (eq2)from eq1 we have x = 45 - yand then substituting this to eq2 we have(45-y)/y +y(45-y) = 2.05that eliminates one unknown leading to the other...^,^that equation is equivalent to:[(45-y)(45-y) + y]/(45-y)y = 2.05next step, is to transpose the denominator to the other side...so we have...(45-y)²+ y = 2.05(45-y)yexpanding it...45² - 90y + y² = 2.05(45y) - 2.05y²simplifyin...2025 - 90y - 92.25y + y² + 2.05y² = 0voila! we have a quadratic equation...3.05y² - 182.25y + 2025 = 0 _______y = [b±√(b² - 4ac)]/2a ___________________y = [182.25±√(182.25² - 4(3.05)(2025)]/2(3.05)y = (182.25±92.25)/6.1y = 45 or y = 14.7545 would be an improbable answer since the sum of the two numbers is 45! get it? it's like this...let's look at eq2 which is -----> x/y + y/x = 2.05if y is equal to 45 then x would be 0 which means that it'll not satisfy the second eq...ok?therefore, y = 14.75 since, x = 45 - ythen, x = 30.25So, basically you are just goin to identify first the two availble equations...then combine it to eliminate one of the two unknowns...and voila! you'll get the answer...hope this will you help you...^-^â€*¬ټ
 
Old 07-30-2007   #4
Shobiz
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This process is much simpler. Please notice the steps and you'll understand what I'm saying.Let the two unknown numbers be 'a' and 'b'.According to the question;1) a+ b = 452) (a/b) + (b/a) = 2.05Solving 2);(a/b) + (b/a) = 2.05=> (a^2 + b^2)/ (ab) =2.05=> (a + b)^2 -2ab = 2.05 (ab)NOTE: a^2 + b^2 = (a+b)^2 -2ab Now, we'll use, 1) a+b = 45;=> (45)^2 -2ab = 2.05ab=> 2025 = 4.05ab=> ab = 2025/4.05 = 500 (This is the product of the two unknown variables).Now, (a-b)^2 = (a+b)^2 -4ab = (45)^2 - 4(500) = 2025 -2000 = 25Therefore, (a-b) = + or - 5 Now, taking case a);a+b =45a-b = 5Soliving the above two equations, we get;a = 25b =20 Taking case b);a+b =45a-b = -5On solving these two equations, we get;a = 20b =25Therefore, the two numbers are 20 and 25 and the product is 500.
 
 

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