We will solve different types of problems on change the subject of a formula.
The subject of a formula is a variable whose relation with other variables of the context is sought and the formula is written in such a way that subject is expressed in terms of the other variables.
For example, in the formula A = \(\frac{1}{2}\)bh, A is the subject which in terms of the other variables b and h.
By knowing the values of the variables b and h, the value of the subject A can be easily calculated. For example, if the base of a triangle is 6 cm and the height is 4 cm, its area
A = \(\frac{1}{2}\)bh = A = \(\frac{1}{2}\) × 6 × 4 cm^{2} = 12 cm^{2}
When a formula involving certain variables is known, we can change the subject of the formula.
Solved examples to change the subject of a formula:
1. In the formula S = \(\frac{n}{2}\)[2a + (n  1) d], S is the subject. Write the formula with d as the subject.
Solution:
Given S = \(\frac{n}{2}\)[2a + (n  1) d]
⟹ 2S = 2an + n(n 1)d
⟹ 2S – 2an = n(n  1)d
⟹ n(n  1)d = 2(S  an)
⟹ d = \(\frac{2(S  an)}{n(n  1)}\). Here, d is the subject.
2. If a = 2b + \(\sqrt{b^{2} + m}\), express m in terms of a and b.
Solution:
Here, a = 2b + \(\sqrt{b^{2} + m}\)
⟹ a  2b = \(\sqrt{b^{2} + m}\)
Squaring the both sides we get,
⟹ (a  2b)^{2} = b^{2} + m
⟹ (a  2b)^{2}  b^{2} = m
⟹ {(a  2b) + b}{(a  2b)  b} = m
⟹ (a  b)(a  3b) = m
⟹ m =(a  b)(a  3b)
3. Make u the subject of the formula f = \(\frac{uv}{u + v}\).
Solution:
Give, f = \(\frac{uv}{u + v}\)
⟹ \(\frac{1}{f}\) = \(\frac{u + v}{uv}\)
⟹ \(\frac{1}{f}\) = \(\frac{1}{u}\) + \(\frac{1}{ v}\)
⟹ \(\frac{1}{u}\) = \(\frac{1}{f}\)  \(\frac{1}{v}\)
⟹ \(\frac{1}{u}\) = \(\frac{v  f}{fv}\)
⟹ u = \(\frac{fv}{v  f}\). Here, u is the subject.
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