Rearranging Formulas

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Notes de leçon

What is a Formula?

A formula is a rule or relationship between quantities, written in shorthand using letters (variables).
Formulas always include an equals sign.
Examples: equation of a straight line y = mx + c, area of a trapezium \text{Area} = \frac{(a+b)h}{2}, Pythagoras' theorem a² + b² = c².

Subject of a Formula

The subject is the variable that stands alone on one side of the equals sign.
In y = mx + c, y is the subject.
To change the subject, rearrange the formula using inverse operations.

Basic Rearrangement Steps

Step 1: Remove any fractions by multiplying both sides by the lowest common denominator.
Step 2: Use inverse operations to isolate the desired variable.
Inverse operations: addition/subtraction, multiplication/division, squares/square roots.
Treat the rearrangement like solving an equation.

Example: Simple Linear Formula

Make x the subject of 5x + 6 = 2y.
Subtract 6: 5x = 2y - 6.
Divide by 5: x = \frac{2y - 6}{5}.
Alternative forms: x = \frac{2}{5}y - \frac{6}{5}, x = 0.4(y - 3), x = 0.4y - 1.2.

Dealing with Brackets

If the variable is inside brackets, either expand the brackets or divide both sides by the coefficient.
Example: Make x the subject of 3(1+x) = y.
Expand: 3 + 3x = y \Rightarrow 3x = y - 3 \Rightarrow x = \frac{y-3}{3}.
Divide first: 1+x = \frac{y}{3} \Rightarrow x = \frac{y}{3} - 1 (equivalent).
If the variable is not inside the bracket, simply divide by the bracket factor.

Fractions within Fractions

When the subject appears in a fraction within a fraction, rewrite using division or multiply numerator and denominator by the common denominator.
Example: x = \dfrac{3/t}{2} becomes x = \dfrac{3}{t} \div 2 = \dfrac{3}{t} \times \dfrac{1}{2} = \dfrac{3}{2t}.
Alternatively, multiply the top and bottom by t: x = \dfrac{\frac{3}{t} \times t}{2 \times t} = \dfrac{3}{2t}.

Dealing with Negative Signs

\frac{a}{-b} = -\frac{a}{b} = \frac{-a}{b}.
Example: -2x = y - 3 \Rightarrow x = \frac{y-3}{-2} = -\frac{y-3}{2} = \frac{3-y}{2}.
Be careful with brackets when moving negative signs.

Worked Examples from Source

Make x the subject of 4m + 5x = 3: 5x = 3 - 4m, x = \frac{3-4m}{5}.
Make x the subject of 3t = \frac{2}{x}: multiply by x: 3tx = 2, divide by 3t: x = \frac{2}{3t}.
Make x the subject of A = \frac{9(1-4x)}{2g}: multiply by 2g: 2gA = 9(1-4x), expand: 2gA = 9 - 36x, rearrange: 36x = 9 - 2gA, x = \frac{9-2gA}{36}.
Alternative forms: \frac{2gA-9}{-36}, -\frac{2gA-9}{36}, \frac{1}{4} - \frac{gA}{18}.

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1.Make x the subject of the formula: 2y = 5x − 7
Easy
- Ax = (2y + 7)/5
- Bx = (2y − 7)/5
- Cx = 2y/5 + 7
2.Make m the subject of the formula: y = 4m − p
Easy
- Am = (y + p)/4
- Bm = (y − p)/4
- Cm = y/4 + p
- Dm = y/4 − p
3.Make r the subject of the formula: p = 3r − 5
Easy
- Ar = (p + 5)/3
- Br = (p − 5)/3
- Cr = p/3 + 5
- Dr = p/3 − 5
4.Make p the subject of the formula: H = 7p − 3
Easy
- Ap = (H + 3)/7
- Bp = (H − 3)/7
- Cp = H/7 + 3
- Dp = H/7 − 3
5.Make x the subject of the formula: T = (1/3)(5x + 2)
Medium
- Ax = (3T − 2)/5
- Bx = (3T + 2)/5
- Cx = (T − 2)/15
- Dx = (T/3 − 2)/5
6.Rearrange the formula 5w − 3y + 7 = 0 to make w the subject.
Medium
- Aw = (3y − 7)/5
- Bw = (3y + 7)/5
- Cw = (−3y + 7)/5
- Dw = (−3y − 7)/5
7.Rearrange T = 5(p + 2) to make p the subject.
Medium
- Ap = T/5 − 2
- Bp = T/5 + 2
- Cp = (T − 2)/5
- Dp = (T + 2)/5
8.Rearrange 2(w + h) = P to make w the subject.
Medium
- Aw = P/2 − h
- Bw = P/2 + h
- Cw = (P − h)/2
- Dw = (P + h)/2

Plus de questions

9.Make r the subject of the formula: T = 3(r + 5)
Medium
- Ar = T/3 − 5
- Br = T/3 + 5
- Cr = (T − 5)/3
- Dr = (T + 5)/3
10.Make y the subject of the equation 5x − 2y + 7 = 0.
Medium
- Ay = (5x + 7)/2
- By = (5x − 7)/2
- Cy = (−5x − 7)/2
- Dy = (−5x + 7)/2
11.Rearrange y = 2x + 1 to make x the subject.
Easy
- Ax = (y − 1)/2
- Bx = (y + 1)/2
- Cx = y/2 − 1
- Dx = y/2 + 1
12.Rearrange this formula to make x the subject: 5x² − 3y = 4y + 8
Hard
- Ax = √((7y + 8)/5)
- Bx = √((7y − 8)/5)
- Cx = √((y + 8)/5)
- Dx = (7y + 8)/5
13.The speed s m/s from a height h metres is given by s = √(19.6h). Make h the subject.
Medium
- Ah = s²/19.6
- Bh = s/19.6
- Ch = √(s)/19.6
- Dh = 19.6/s²
14.A = (2π + y)x². Rearrange to make x the subject.
Medium
- Ax = √(A/(2π + y))
- Bx = A/(2π + y)
- Cx = √(A) − (2π + y)
- Dx = √(A/(2π)) + y
15.m = 2p + x/y. Make x the subject.
Medium
- Ax = y(m − 2p)
- Bx = y(m + 2p)
- Cx = (m − 2p)/y
- Dx = m/y − 2p
16.Make y the subject of the formula h² = x² + 2y².
Medium
- Ay = √((h² − x²)/2)
- By = √(h² − x²)/2
- Cy = (h² − x²)/2
- Dy = √(h² − x²)
17.Make x the subject of the formula: y = 3x + 2
Easy
- Ax = (y - 2)/3
- Bx = (y + 2)/3
- Cx = y/3 - 2
- Dx = 3y - 2
18.Make t the subject of the formula: v = u + at
Easy
- At = (v - u)/a
- Bt = v - u - a
- Ct = (v + u)/a
- Dt = (v - u)a
19.Make b the subject of the formula: A = 1/2 × b × h
Easy
- Ab = 2A/h
- Bb = A/(2h)
- Cb = 2Ah
- Db = A/2 - h
20.Make x the subject of the formula: 3x + 2y = 7
Medium
- Ax = (7 - 2y)/3
- Bx = (7 + 2y)/3
- Cx = 7 - 2y - 3
- Dx = (2y - 7)/3
21.Make r the subject of the formula: C = 2πr
Easy
- Ar = C/(2π)
- Br = 2πC
- Cr = C - 2π
- Dr = C/2 - π
22.Make x the subject of the formula: y = (x + 3)/4
Medium
- Ax = 4y - 3
- Bx = 4y + 3
- Cx = (y - 3)/4
- Dx = y/4 + 3
23.Make a the subject of the formula: v² = u² + 2as
Medium
- Aa = (v² - u²)/(2s)
- Ba = (v² + u²)/(2s)
- Ca = v² - u² - 2s
- Da = (v² - u²)/s
24.Make x the subject of the formula: y = (2x + 1)/3
Medium
- Ax = (3y - 1)/2
- Bx = (3y + 1)/2
- Cx = 3y/2 - 1
- Dx = (y - 1)/6
25.Make t the subject of the formula: s = 1/2 × g × t²
Medium
- At = √(2s/g)
- Bt = √(s/(2g))
- Ct = 2s/g
- Dt = √(2sg)
26.Make y the subject of the formula: x = (y + 2)/(y - 3)
Hard
- Ay = (3x + 2)/(x - 1)
- By = (3x - 2)/(x + 1)
- Cy = (2x + 3)/(x - 1)
- Dy = (2x - 3)/(x + 1)
27.Make L the subject of the formula: P = 2L + 2W
Easy
- AL = (P - 2W)/2
- BL = P - 2W
- CL = (P + 2W)/2
- DL = P/2 - W
28.Make d the subject of the formula: C = πd
Easy
- Ad = C/π
- Bd = Cπ
- Cd = C - π
- Dd = C/π + 1
29.Make x the subject of the formula: y = 5 - 2x
Easy
- Ax = (5 - y)/2
- Bx = (y - 5)/2
- Cx = 5 - 2y
- Dx = (5 + y)/2
30.Make x the subject of the formula: y = (x² + 1)/3
Medium
- Ax = √(3y - 1)
- Bx = √(3y + 1)
- Cx = √(y/3 - 1)
- Dx = 3y - 1
31.Make r the subject of the formula: V = 4/3 × π × r³
Medium
- Ar = cube_root(3V/(4π))
- Br = cube_root(4V/(3π))
- Cr = (3V/(4π))^(1/3)
- Dr = (4V/(3π))^(1/3)
32.Make t the subject of the formula: d = rt
Easy
- At = d/r
- Bt = dr
- Ct = d - r
- Dt = r/d
33.A rectangle has length L and width W. Its perimeter P is given by P = 2L + 2W. Which expression correctly gives L in terms of P and W?
Easy
- AL = (P - 2W)/2
- BL = P - 2W
- CL = (P - W)/2
- DL = P/2 - W
34.The formula for the volume of a cone is V = (1/3)πr²h. Make h the subject.
Medium
- Ah = 3V/(πr²)
- Bh = V/(3πr²)
- Ch = 3Vπ/r²
- Dh = (3V)/(πr²)
35.A car travels at speed v for time t, covering distance d. The formula is d = vt. Make t the subject.
Easy
- At = d/v
- Bt = v/d
- Ct = dv
- Dt = d - v
36.The formula for the surface area of a sphere is A = 4πr². Make r the subject.
Medium
- Ar = √(A/(4π))
- Br = A/(4π)
- Cr = √(A/4π)
- Dr = A/(2π)
37.The formula for converting temperature from Celsius (C) to Fahrenheit (F) is F = (9/5)C + 32. Make C the subject.
Medium
- AC = (5/9)(F - 32)
- BC = (9/5)(F - 32)
- CC = (F - 32)/ (9/5)
- DC = (5/9)F - 32
38.Given the formula y = (x + 2)/(x - 1), make x the subject.
Hard
- Ax = (y + 2)/(y - 1)
- Bx = (2y + 1)/(y - 1)
- Cx = (y - 2)/(y + 1)
39.The formula for the period of a pendulum is T = 2π √(L/g). Make L the subject.
Medium
- AL = (T² g)/(4π²)
- BL = (T g)/(2π)
- CL = (T²)/(4π² g)
- DL = (2πT)/g
40.Make x the subject of the formula: y = (3x - 5)/2.
Medium
- Ax = (2y + 5)/3
- Bx = (2y - 5)/3
- Cx = (2y + 5)/2
- Dx = (2y - 5)/2
41.In the formula y = mx + c, the subject is x.
Easy
True or false?
42.To make x the subject of y = 3x + 2, the first step is to:
Easy
- AAdd 2 to both sides
- BSubtract 2 from both sides
- CDivide both sides by 3
- DMultiply both sides by 3
43.Make b the subject of A = 1/2 × b × h.
Easy
- Ab = A / (2h)
- Bb = 2A / h
- Cb = 2Ah
44.Rearranging y = 5 − 2x to make x the subject gives x = (y − 5)/2.
Easy
True or false?
45.Make x the subject of y = (x² + 1)/3.
Medium
- Ax = √(3y − 1)
- Bx = √(3y) − 1
- Cx = √(3y + 1)
- Dx = √(3y) + 1
46.Match each formula to its correct rearrangement to make the variable shown in brackets.
Medium
- F = ma (m)
- A = πr² (r)
- P = 2L + 2W (L)
- m = F/a
- r = √(A/π)
- L = (P − 2W)/2
47.Make x the subject of the formula: y = 5x + 2
Easy
- Ax = (y - 2)/5
- Bx = (y + 2)/5
- Cx = y/5 - 2
- Dx = y/5 + 2
48.To make x the subject of y = 3x + 2, the first step is to subtract 2 from both sides.
Easy
True or false?
49.Order these steps to make x the subject of y = (2x + 1)/3.
Medium
- Multiply both sides by 3: 3y = 2x + 1
- Subtract 1: 3y - 1 = 2x
- Divide by 2: x = (3y - 1)/2
50.Make x the subject of the formula: y = (3x - 5)/2
Medium
- Ax = (2y + 5)/3
- Bx = (2y - 5)/3
- Cx = 2y + 5/3
- Dx = 3y + 5/2
51.Given the formula P = 2L + 2W, find L when P = 30 and W = 5.
Easy
52.Complete the sentence.
Easy
To make a the subject of the formula v = u + at, first subtract ____ from both sides, then divide by ____.
53.Rearrange the formula v = u + at to make a the subject.
Medium
54.Make d the subject of C = πd. Then find d when C = 31.42 (use π = 3.142).
Medium
55.Complete the rearrangement.
Medium
Make x the subject of y = (2x + 1)/3. x = (____ - 1)/____
56.Complete the rearrangement:
Medium
Make r the subject of V = (4/3)πr³. r = ____
57.Make y the subject of the formula: x = (y + 2)/(y - 3).
Hard
58.Make t the subject of the formula: v = u + at. Then find t when v = 20, u = 5, and a = 3.
Medium
59.Make x the subject of the formula: y = (x + 2)/(x - 3)
Hard
60.Given the formula P = 2L + 2W, find L when P = 30 and W = 5.
Easy
- A10
- B12
- C11
- D9
61.Make d the subject of C = πd. Then find d when C = 31.42 (use π = 3.142).
Medium
- A9
- B11
- C12
- D10
62.Make t the subject of the formula: v = u + at. Then find t when v = 20, u = 5, and a = 3.
Medium
- A5
- B4
- C6
- D7

Notes de révision

## Rearranging Formulas (Changing the Subject) - **Subject**: The variable alone on one side of the equals sign. E.g., in y = mx + c, y is the subject. - **Goal**: Isolate the desired variable using inverse operations. ### Steps: 1. **Remove fractions**: Multiply both sides by the denominator. 2. **Expand brackets** only if the target variable is inside them; otherwise, divide by the bracket. 3. **Use inverse operations** (add/subtract, multiply/divide) to isolate the variable. 4. **Simplify**: Combine like terms, factor if needed. ### Examples: - Make x the subject of 2y = 5x − 7: Add 7: 2y + 7 = 5x → Divide by 5: x = (2y + 7)/5 - Make m the subject of y = 4m − p: Add p: y + p = 4m → Divide by 4: m = (y + p)/4 - Make r the subject of p = 3r − 5: Add 5: p + 5 = 3r → Divide by 3: r = (p + 5)/3 - Make p the subject of H = 7p − 3: Add 3: H + 3 = 7p → Divide by 7: p = (H + 3)/7 - Make x the subject of T = 1/3 (5x + 2): Multiply by 3: 3T = 5x + 2 → Subtract 2: 5x = 3T − 2 → Divide by 5: x = (3T − 2)/5 - Make w the subject of 5w – 3y + 7 = 0: Add 3y − 7: 5w = 3y − 7 → Divide by 5: w = (3y − 7)/5 - Make p the subject of T = 5(p + 2): Divide by 5: T/5 = p + 2 → Subtract 2: p = T/5 − 2 - Make w the subject of 2(w + h) = P: Divide by 2: w + h = P/2 → Subtract h: w = P/2 − h - Make r the subject of T = 3(r + 5): Divide by 3: T/3 = r + 5 → Subtract 5: r = T/3 − 5 - Make y the subject of 5x − 2y + 7 = 0: Subtract 5x + 7: −2y = −5x − 7 → Multiply by −1: 2y = 5x + 7 → Divide by 2: y = (5x + 7)/2 - Make x the subject of y = 2x + 1: Subtract 1: y − 1 = 2x → Divide by 2: x = (y − 1)/2 - Make x the subject of 5x² − 3y = 4y + 8: Add 3y: 5x² = 7y + 8 → Divide by 5: x² = (7y + 8)/5 → Square root: x = ±√[(7y + 8)/5] - Make h the subject of s = √(19.6h): Square: s² = 19.6h → Divide by 19.6: h = s²/19.6 - Make x the subject of A = (2π + y)x²: Divide by (2π + y): x² = A/(2π + y) → Square root: x = ±√[A/(2π + y)] - Make x the subject of m = 2p + x/y: Subtract 2p: m − 2p = x/y → Multiply by y: x = y(m − 2p) - Make y the subject of h² = x² + 2y²: Subtract x²: 2y² = h² − x² → Divide by 2: y² = (h² − x²)/2 → Square root: y = ±√[(h² − x²)/2] ### Handling Negatives: - a/−b = −a/b = −(a/b) - −a/−b = a/b ### Fractions in Fractions: - Rewrite as division: x = (3/t)/2 = 3/(2t) - Multiply top and bottom by LCD. ### Tips: - If the variable is not inside a bracket, do not expand. - Different forms of the answer are acceptable if simplified.

Subject: The variable that stands alone on one side of the equals sign in a formula.
Rearrange: To change the subject of a formula by using inverse operations.
Inverse operation: An operation that undoes another (e.g., addition undoes subtraction, multiplication undoes division).
Formula: A rule or relationship between quantities, written with variables and an equals sign.

Cartes mémo

What does it mean to make a variable the subject of a formula?

The variable is on its own on one side of the equals sign. For example, y is the subject of y = mx + c.

Make x the subject: 2y = 5x − 7

x = (2y + 7)/5

Make m the subject: y = 4m − p

m = (y + p)/4

Make r the subject: p = 3r − 5

r = (p + 5)/3

Make p the subject: H = 7p − 3

p = (H + 3)/7

Make x the subject: T = 1/3 (5x + 2)

x = (3T − 2)/5

Rearrange 5w – 3y + 7 = 0 to make w the subject.

w = (3y − 7)/5

Rearrange T = 5(p + 2) to make p the subject.

p = T/5 − 2

Rearrange 2(w + h) = P to make w the subject.

w = P/2 − h

Make r the subject: T = 3(r + 5)

r = T/3 − 5

Make y the subject: 5x − 2y + 7 = 0

y = (5x + 7)/2

Rearrange y = 2x + 1 to make x the subject.

x = (y − 1)/2

Make x the subject: 5x² − 3y = 4y + 8

x = √((7y + 8)/5)

Given s = √(19.6h), make h the subject.

h = s²/19.6

Make x the subject: A = (2π + y)x²

x = √(A/(2π + y))

Make x the subject: m = 2p + x/y

x = y(m − 2p)

Make y the subject: h² = x² + 2y²

y = √((h² − x²)/2)

How do you remove fractions when rearranging?

Multiply both sides by the lowest common denominator.

If the variable is inside brackets, what two methods can you use?

Expand the brackets, or divide both sides by the coefficient outside the brackets.

How do you simplify a fraction within a fraction like x = (3/t)/2?

Rewrite as division: x = 3/t ÷ 2 = 3/(2t).

What is the result of dividing by a negative? For example, −2x = y − 3.

x = (y − 3)/(−2) which equals −(y − 3)/2 or (3 − y)/2.

True or false: If the variable you want is not inside a bracket, you must expand the bracket.

False. You can simply divide both sides by the bracket.

Make x the subject: 4m + 5x = 3

x = (3 − 4m)/5

Make x the subject: 3t = 2/x

x = 2/(3t)

Make x the subject: A = 9(1 − 4x)/(2g)

x = (9 − 2gA)/36

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