One positive integer is 1 less than twice another. The sum of their squares is 106. Find the integers.
1 answer:
They give us 2 pieces to the puzzle. Both are positive numbers...x and y. 1.) 1 number is 1 less than twice another number. (x = 2y -1)...and 2.) the sum of their squares is 106. (x^2 + y^2 = 106). substitute the value for x into the second equation. (2y-1)^2 + y^2 = 106 (2y-1) (2y-1) + y^2 = 106 (use distributive property) 4y^2 - 2y - 2y + 1 + y^2 = 106 (subtract 106 from both sides) 4y^2 - 2y - 2y + 1 + y^2 - 106 = 106 - 106 (combine like terms) 5y^2 - 4y - 105 = 0 (factor) (y-5) (5y-21) = 0 (set to 0) y - 5 = 0 y = 5 substitute the 5 into the equation for y (x = 2(5) - 1) x = 9 if we square 9, we get 81. subtracted from 106 we have 25...the square root of 25 is 5. our answers are 5 and 9.
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