Linear solutions

Algebra Level pending

For a certain value of a a and b b the following pair of linear equations have an infinite number of solutions:

2 x + 3 y = 7 2x+3y=7

( a − b ) x + ( a + b ) y = 3 a + b − 2 (a-b)x+(a+b)y=3a+b-2

Find a + b a × b \frac{a+b}{a×b}


The answer is 1.2.

Sahar Bano by Sahar Bano , 38, India

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1 solution

The given equations will have an infinite number of solutions if

a − b 2 = a + b 3 = 3 a + b − 2 7 = k \dfrac{a-b}{2}=\dfrac{a+b}{3}=\dfrac{3a+b-2}{7}=k (say).

Then, a − b = 2 k , a + b = 3 k a-b=2k, a+b=3k

⟹ a = 5 k 2 , b = k 2 , 3 a + b − 2 = 7 k \implies a=\dfrac{5k}{2}, b=\dfrac{k}{2}, 3a+b-2=7k

⟹ 15 k 2 + k 2 − 2 = 7 k ⟹ k = 2 \implies \dfrac{15k}{2}+\dfrac{k}{2}-2=7k\implies k=2 .

Therefore a = 5 , b = 1 , a=5, b=1, and a + b a b = 5 + 1 5 × 1 = 1.2 \dfrac{a+b}{ab}=\dfrac{5+1}{5\times 1}=\boxed {1.2} .

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