SOLUTION: One number is 3 more than another number. The sum of their reciprocals is 1/2. what are the numbers?

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Question 431489: One number is 3 more than another number. The sum of their reciprocals is 1/2. what are the numbers?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Call the numbers a and b
given:
(1) +a+=+b+%2B+3+
(2) +1%2Fa+%2B+1%2Fb+=+1%2F2+
---------------------
Multiply both sides of (2) by 2a%2Ab
(2) +2b+%2B+2a+=+a%2Ab+
By substitution:
(2) +2b+%2B+2%2A%28b+%2B+3%29+=+%28b+%2B+3%29%2Ab+
(2) +2b+%2B+2b+%2B+6+=+b%5E2+%2B+3b+
(2) +b%5E2+-+b+=+6+
Complete the square
+b%5E2+-+b+%2B+%281%2F2%29%5E2+=+6+%2B+%281%2F2%29%5E2+
+%28+b+-+1%2F2%29%5E2+=+24%2F4+%2B+1%2F4+
+%28+b+-+1%2F2%29%5E2+=+25%2F4+
Take the square root of both sides
+b+-+1%2F2+=+5%2F2+
+b+=+3+ and also
+b+-+1%2F2+=+-5%2F2+
+b+=+-2+
And, from (1),
(1) +a+=+-2+%2B+3+
+a+=+1+
The numbers are 1 and -2
check answer:
(2) +1%2Fa+%2B+1%2Fb+=+1%2F2+
(2) +1%2F1+%2B+1%2F%28-2%29+=+1%2F2+
+1+-+1%2F2+=+1%2F2+
OK