Suppose y varies as the sum of two quantities, one of which varies
directly as x and the other inversely with x. When x = 2, y=4 and when
x = 5 , y = 7. Find the relationship between x and y.
Solution.
Let the two quantities be a and b.
Then,
y= k (a + b ) ...........................................(1).
a= mx .......................................................(2)
b = n/x ......................................................(3).
K, m and n are the constants of variation. Using the same letter for
the 3 constants lead to a deadlock.
Substitute Equations 2 and 3 into Equation 1.
y = kmx + (kn /x) ..................................(4).
x =2 when y= 4 turns equation 4 into:
4 = 2mk + (kn/2).
8 = 4mk + kn ...................................(5).
x=5 when y =7 turns equation 4 into :
7 = 5mk + (kn /5).
35 = 25mk + kn .............................(6).
Solve equations 5 and 6 simultaneously.
km = 9/7 and kn = 20/7.
Hence, equation 4 becomes :
y = (9/7)x + [ (20/7) ÷ x ]
= (9x/7) + (20/7x)
= (9x² + 20)/(7x).
This is the required relationship.
Confirm that y =4 when x =2 and y =7 when x =5 by substituting 2 for x
and later 5 for x in the derived relationship.
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