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# --- 2 Quick Maths Questions --- watch

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1. Solve the simultaneous equations:

x + y = 1

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

What is the value of xy?

And this one..

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

2. 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
=============
3. 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
4. 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.
5. You two are top blokes

Thanks for trying Fermat..i got that too, but the teacher said it was wrong
6. 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.
7. 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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Updated: January 3, 2005
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