Rearranging Formulas

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課程筆記

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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練習題

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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

複習筆記

## 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.

字卡

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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