(a) Make q the subject of the relation (t = sqrt{frac{pq}{r} – r^{2}q}).
(b) If (9^{(1 – x)} = 27^{y}) and (x – y = -1frac{1}{2}), find the value of x and y.
Explanation
(a) (t = sqrt{frac{pq}{r} – r^{2}q})
Squaring both sides,
(t^{2} = frac{pq}{r} – r^{2}q)
(t^{2} = frac{pq – r^{3}q}{r})
(t^{2}r = pq – r^{3}q implies t^{2}r = q(p – r^{3}))
(q = frac{t^{2}r}{p – r^{3}}).
(b) (9^{(1 – x)} = 27^{y})
(implies 3^{2(1 – x)} = 3^{3y})
(2 – 2x = 3y ….. (1))
(x – y = -1frac{1}{2} ……. (2))
From (2), (x = -1frac{1}{2} + y)
(therefore 2 – 2(-1frac{1}{2} + y) = 3y)
(2 + 3 – 2y = 3y implies 5 = 3y + 2y = 5y)
(implies y = 1)
(x = -1frac{1}{2} + 1 = -frac{1}{2})
(therefore x + y = -frac{1}{2} + 1 = frac{1}{2})