Change of Subject of Formula – JSS3 Mathematics Lesson Note (Third Term)

Changing the subject of a formula means rearranging it so that a different variable stands alone on one side of the equals sign.

Behavioural objectives

By the end of the lesson, students should be able to:

1. State what a formula is and what is meant by its subject.
2. Substitute given values into a formula.
3. Change the subject of a formula using inverse operations.
4. Find the value of a variable after changing the subject.

What is a formula?

A formula is a general equation involving two or more unknowns.

• Area of a circle: a = πr², which gives the area a in terms of the radius r.
• Simple interest: I = PRT, where P is the principal, R the rate and T the time.
• Velocity: v = u + at, where u is the starting speed, a the acceleration and t the time.
• Volume of a cylinder: V = πr²h, where r is the radius and h the height.

What is the subject of a formula?

The subject of a formula is the variable that stands alone on one side of the equals sign.

• In a = πr²: a is the subject.
• In I = PRT: I is the subject.
• In v = u + at: v is the subject.
• Variable: a letter whose value can change, such as r, h or t.
• Constant: a fixed number, such as 2 or π.

How do you change the subject of a formula?

To change the subject, use inverse operations on both sides until the chosen variable stands alone.

What are inverse operations?

An inverse operation is the operation that undoes another operation.

What are the steps?

1. Clear any fractions by cross multiplying or multiplying both sides by the denominator.
2. Expand any brackets that contain the new subject.
3. Collect all terms containing the new subject on one side.
4. Move all other terms to the opposite side.
5. If the new subject appears in more than one term, factorise it out.
6. Divide both sides by the factor multiplying the new subject.
7. If the subject is squared, take the square root of both sides.

How do you change the subject? Worked examples

Example 1

In the formula S = 2πr(r + h), make h the subject.

1. Expand the bracket: S = 2πr² + 2πrh
2. Subtract 2πr² from both sides: S – 2πr² = 2πrh
3. Divide both sides by 2πr: h = (S – 2πr²)/(2πr)

Example 2

Make m the subject if F = (mv – mu)/t.

1. Cross multiply: Ft = mv – mu
2. Factorise the right side: Ft = m(v – u)
3. Divide both sides by (v – u): m = Ft/(v – u)

Example 3

Make U the subject if V² = U² + 2as.

1. Subtract 2as from both sides: V² – 2as = U²
2. Take the square root of both sides: U = √(V² – 2as)

Example 4

Make h the subject, then make a the subject, if A = ½h(a + b).

1. Multiply both sides by 2: 2A = h(a + b)
2. Divide both sides by (a + b): h = 2A/(a + b)
3. To make a the subject, divide 2A = h(a + b) by h: 2A/h = a + b
4. Subtract b from both sides: a = 2A/h – b

How do you find the value of a variable after changing the subject?

Change the subject first, then substitute the given values into the new formula.

Example

Given mx + c = y, express x in terms of m, c and y. Find the value of x if y = 10, c = 2 and m = 4.

1. Write the formula: mx + c = y
2. Subtract c from both sides: mx = y – c
3. Divide both sides by m: x = (y – c)/m
4. Substitute y = 10, c = 2 and m = 4: x = (10 – 2)/4
5. Simplify: x = 8/4 = 2

• Check: mx + c = 4(2) + 2 = 10, which equals y.

What mistakes should you avoid?

• Changing one side only: whatever you do to one side, do to the whole of the other side.
• Dividing only part of an expression: divide every term on the side, or use brackets.
• Forgetting to factorise: if m appears twice, as in mv – mu, factorise before dividing.
• Square root of one term: take the square root of the whole side, as in √(V² – 2as).
• Sign errors: a term changes sign when it moves to the other side.

Evaluation

1. If I = PRT, make P the subject and find the value of P if I = 10, R = 4 and T = 5.
2. Make r the subject if V = πr²h.
3. Make u the subject if 1/f = 1/v + 1/u.

Assignment

Reading assignment

1. Exam focus for J.S.C.E, pages 212 to 216.
2. Essential Maths for J.S.S.3, pages 47 to 54.

Weekend assignment

1. Express a in terms of u, v and t in v = u + at.
   (a) a = vu – t (b) a = (v – u)/t (c) a = (v + u)/t
2. If Z = 2p + 3, find the value of Z when p = 1.
   (a) Z = 2 (b) Z = 5 (c) Z = 7
3. In the relation X = m – 6y, how would you write m in terms of x and y?
   (a) m = x – 6 (b) m = 6x – 6y (c) m = 6x + 6y
4. Express n in terms of S, a and L if S = (n/2)(a + L).
   (a) n = 2(a + L) (b) n = 2S/(a + L) (c) 2S – (a – L) (d) S/(a + L)
5. Make U the subject if V² = U² + 2as.
   (a) U = V² – 2as (b) U = √(V² – 2as) (c) V² – 2a

Theory

1. Given L = arⁿ⁻¹, make a the subject of the formula.
2. If S + 2t, make t the subject of the formula. Find the value of t if S = 1 and d = 2.