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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