# If 1/x + 1/y = 1/3; 1/x + 1/z = 1/5; 1/y + 1/z = 1/7 then x/y = ?? If 1/x + 1/y = 1/3 1/x + 1/z = 1/5 1/y + 1/z = 1/7 then x/y = ??

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Let 1/x=p, 1/y=q, 1/z=r

the equations are:

p + q = 1/3 --------(1)

p + r = 1/5 --------(2)

q + r = 1/7 --------(3)

subtracting (3) from (2)

p - q = 1/5 - 1/7 = (7-5)/35 = 2/35 ---------(4)

adding (1) and (4)

2p = 1/3 + 2/35 = (35+2×3)/105 = 41/105

⇒ p = 41/210

⇒ x = 1/p = 210/41

Putting p in equation (1)

41/210 + q = 1/3

⇒ q = 1/3 - 41/210 = (70-41)/210

⇒ q = 29/210

⇒ y = 1/q = 210/29

x/y = (210/41) / (210/29) = (210/41) × (29/210) = 29/41

the equations are:

p + q = 1/3 --------(1)

p + r = 1/5 --------(2)

q + r = 1/7 --------(3)

subtracting (3) from (2)

p - q = 1/5 - 1/7 = (7-5)/35 = 2/35 ---------(4)

adding (1) and (4)

2p = 1/3 + 2/35 = (35+2×3)/105 = 41/105

⇒ p = 41/210

⇒ x = 1/p = 210/41

Putting p in equation (1)

41/210 + q = 1/3

⇒ q = 1/3 - 41/210 = (70-41)/210

⇒ q = 29/210

⇒ y = 1/q = 210/29

x/y = (210/41) / (210/29) = (210/41) × (29/210) = 29/41