SOLUTION: The sum of the ages of two brothers is 22 years. Six years ago, the product of their ages was 21 years. Find the age of the elder brother.

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Question 754338: The sum of the ages of two brothers is 22 years. Six years ago, the product of their ages was 21 years. Find the age of the elder brother.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +a+ = the age of the elder brother
Let +b+ = the age of the younger brother
given:
(1) +a+%2B+b+=+22+
(2) +%28+a+-+6+%29%2A%28+b+-+6+%29+=+21+
--------------------------
(1) +a+%2B+b+=+21+%2B+1+
(1) +21+=+a+%2B+b+-+1+
Substitute this result into (2)
(2) +%28+a+-+6+%29%2A%28+b+-+6+%29+=+a+%2B+b+-+1+
(2) +a%2Ab+-+6b+-+6a+%2B+36+=+a+%2B+b+-+1+
(2) +a%2Ab+-+6b+-+b+-+6a+-+a+%2B+37+=+0+
(2) +a%2Ab+-+7a+-+7b+%2B+37+=+0+
and, since
+b+=+-a+%2B+22+
(2) +a%2A%28+-a+%2B+22+%29+-+7a+-+7%2A%28+-a+%2B+22+%29+%2B+37+=+0+
(2) +-a%5E2+%2B+22a+-+7a+%2B+7a+-+154+%2B+37+=+0+
(2) +-a%5E2+%2B+22a+-117+=+0+
Use quadratic formula
+a+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+%28-22+%2B-+sqrt%28+22%5E2+-+4%2A%28-1%29%2A%28-117%29+%29%29+%2F+%282%2A%28-1%29%29+
+a+=+%28-22+%2B-+sqrt%28+484+-+468+%29%29+%2F+%28-2%29+
+a+=+%28-22+%2B-+sqrt%28+16+%29%29+%2F+%28-2%29+
+a+=+%28+-22+%2B+4+%29+%2F+%28-2%29+
+a+=+-18%2F%28-2%29+
+a+=+9+
-------------
The other answer is:
+a+=+%28+-22+-+4+%29+%2F+%28-2%29+
+a+=+-26+%2F+%28-2%29+
+a+=+13+
I'll take the larger number since +a+ is the elder brother
+b+=+-a+%2B+22+
+b+=+-13+%2B+22+
+b+=+9+
The elder brother is 13
check:
(2) +%28+a+-+6+%29%2A%28+b+-+6+%29+=+21+
(2) +%28+13+-+6+%29%2A%28+9+-+6+%29+=+21+
(2) +7%2A3+=+21+
(2) +21+=+21+
OK