Solve for y

Algebra Level 4

Given that y y is a real number such that

1 − y − 3 y + 1 5 3 = y 2 − 2 y − 10 − 6 y 7 2 , 1-\frac{y-\frac{3y+1}{5}}{3}=\frac{y}{2}-\frac{2y-\frac{10-6y}{7}}{2},

and y y can be expressed in the form − a b -\frac{a}{b} , where a a and b b are coprime, positive integers, find the value of a + b a+b .


The answer is 241.

Victor Loh by Victor Loh , 19, Singapore

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1 solution

Aaaaaa Bbbbbb
Jul 15, 2014

⇔ ( 10 − 13 y ) / 14 = ( 16 − 2 y ) / 15 \Leftrightarrow (10-13y)/14=(16-2y)/15 ⇔ 10 14 − 6 15 = 13 y 14 − 2 y 15 = ( 15 ∗ 13 − 14 ∗ 2 ) y 14 ∗ 15 \Leftrightarrow \frac{10}{14}-\frac{6}{15}=\frac{13y}{14}-\frac{2y}{15}=\frac{(15*13-14*2)y}{14*15} ⇔ 10 ∗ 15 − 14 ∗ 16 = ( 15 ∗ 13 − 14 ∗ 2 ) y \Leftrightarrow 10*15-14*16=(15*13-14*2)y − 74 y = 167 -74y=167 y = − 74 167 y=\frac{-74}{167} a + b = 241 a+b=\boxed{241}

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