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--- 2 Quick Maths Questions --- watch

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    Solve the simultaneous equations:

    x + y = 1

    x2 + y2 = 2 <--- x2 = x squared.. y2 = y squared

    giving exact, simplified answers.
    What is the value of xy?


    And this one.. :hmpf:

    Find the sum of all even 3 digit numbers less than 1000, which are not multiples of 7.

    Please could you help?! :eek:
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    x = 1-y
    x² + y² = 2
    (1-y)² + y² = 2
    1 - 2y + y² + y² = 2
    2y² - 2y - 1 = 0
    y = [2 +/- rt(4 - 4.2.-1)]/2.2
    y = [2 +/- rt(12)]/4
    y = 1/2 +/- rt(3/4)
    y = -0.366, y = 1.366
    ================
    x = 1.366, x = -0.366
    =================

    In surd form,

    y = ½ +/- √(3)/2
    x= ½ -/+ √(3)/2
    =============
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    formula for 3 digit even numbers is 98+2n. therefore:

    first three digit even number is 98+2(1) = 98+2 = 100
    number of three digit even number is 98+2n < 1000, 902/2 > n, 451 > n, n = 450
    last three digit number is 98+2(450) = 998

    formula for 3 digit even multiples of 7 is 98+14n. therefore:
    number of three digit even number is 98+14n < 1000, 902/14 > n, 64.4..> n, n = 64
    last three digit number is 98+14(64) = 994

    Sum_[even 3 digit] = (n/2)(a+l) = (450/2)(100+998) = 225*1098 = 247050
    Sum_[even 3 digit multiples of 7] = (n/2)(a+l) = (64/2)(112+994) = 32*1106 = 35392

    Sum_[even 3 digit non multiples of 7] = Sum_[even 3 digit] - Sum_[even 3 digit multiples of 7] = 247050-35392 = 211658

    Sum_[even 3 digit non multiples of 7] = 211658
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    x+y=1 (i)
    x²+y²=2 (ii)

    (i)=>y=1-x

    (ii)(i)=>x²+(1-x)²=2
    =>x²+1-2x+x²=2
    =>2x²-2x-1=0

    x=(((-b)+/-√(b²-4ac))/2a)=((2+/-√12)/4)=((2+/-√4√3)/4)=((2+/-2√3)/4)

    Therefore:

    x=(1/2)(1+√3),y=(1/2)(1-√3) OR x=(1/2)(1-√3),y=(1/2)(1+√3)

    xy1=(1/2)(1+√3)*(1/2)(1-√3)=(1/4)(1+√3)(1-√3)=(1/4)(1-√3+√3-3)=(-1/2)

    xy2=(-1/2)

    Therefore:

    xy=(-1/2)

    Newton.
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    You two are top blokes

    Thanks for trying Fermat..i got that too, but the teacher said it was wrong
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    Fermat got the right values of x and y (only he might like to check the sign on one of his values of x). If all you did was state values of x and y then it would be wrong as it asks for xy which Newton has found.
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    Yeah, I had a sign wrong on one of the values - fixed it now - ta.

    The q asked "Solve the simultaneous equations:", which usually means get the values of the unknowns.

    "giving exact, simplified answers." means the answer should be in surd form.

    "What is the value of xy?", this looks like a 2nd part to the question.
 
 
 
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