Rearranging Formula

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課程筆記

Introduction to Rearranging Formulas

A formula is a rule or relationship between quantities, written using variables and an equals sign.
The subject of a formula is the variable on its own on one side (e.g., y is the subject of y = mx + c).
To change the subject, rearrange the formula using inverse operations, similar to solving equations.
First remove any fractions by multiplying both sides by the lowest common denominator.
Then use inverse operations (addition/subtraction, multiplication/division, powers/roots) to isolate the desired variable.

Subject Appears Once: Basic Operations

Use inverse operations step by step. For example, to make x the subject of 2y = 5x − 7: add 7 → 2y + 7 = 5x, then divide by 5 → x = (2y + 7)/5.
If the variable is inside brackets, you can either expand the brackets or divide by the coefficient outside. E.g., 3(1 + x) = y → x = y/3 − 1.
When dealing with fractions in fractions, rewrite using division or multiply numerator and denominator by the common denominator.
If dividing by a negative, remember that a/−b = −a/b = −(a/b). For example, −2x = y − 3 gives x = (3 − y)/2.

Subject Appears Once: Examples with Fractions and Brackets

Example: Make x the subject of 4m + 5x = 3 → subtract 4m: 5x = 3 − 4m, divide by 5: x = (3 − 4m)/5.
Example: Make x the subject of 3t = 2/x → multiply by x: 3tx = 2, divide by 3t: x = 2/(3t).
Example: Make x the subject of A = 9(1 − 4x)/(2g) → multiply by 2g: 2gA = 9(1 − 4x), expand: 2gA = 9 − 36x, then isolate x: x = (9 − 2gA)/36 or equivalent forms.
If the variable is not inside a bracket, you do not need to expand. E.g., (1 + k)x = y → x = y/(1 + k).

Subject Appears Twice: Factorising

When the subject appears twice, collect all terms containing the subject on one side, then factorise to make it appear once.
Example: Make x the subject of x + xy = 3 − 2y → factorise x(1 + y) = 3 − 2y, then divide: x = (3 − 2y)/(1 + y).
If the subject is inside brackets, expand first. E.g., c(x + 2) − x = f → expand: cx + 2c − x = f, then collect x terms: cx − x = f − 2c, factorise: x(c − 1) = f − 2c, so x = (f − 2c)/(c − 1).
If the subject appears on both sides of the equation, bring those terms to the same side before factorising. E.g., 3x = y − px → add px: 3x + px = y, factorise: x(3 + p) = y, so x = y/(3 + p).

Subject Appears Twice: Powers and Roots

If the subject appears with the same power, collect terms and factorise the power. Then apply the inverse root.
Example: Make x the subject of x² = −px² + r → add px²: x² + px² = r, factorise: x²(1 + p) = r, so x² = r/(1 + p), then x = ±√(r/(1 + p)).
When taking roots, remember to apply to the entire expression. E.g., x³ = (t³ + 1)/(t³ + 8) → x = ∛((t³ + 1)/(t³ + 8)).
Be careful: ∛((t³ + 1)/(t³ + 8)) is not equal to (t + 1)/(t + 2).

Subject Appears Twice: Fractional Equations

When the subject appears in a denominator, multiply both sides by the denominator to eliminate the fraction.
Example: Make x the subject of p = (2 − ax)/(x − b) → multiply: p(x − b) = 2 − ax, expand: px − pb = 2 − ax, bring x terms together: px + ax = 2 + pb, factorise: x(p + a) = 2 + pb, so x = (2 + pb)/(p + a).
Always check that you have not lost any solutions, especially when dividing by an expression that could be zero.

Common Mistakes and Tips

Do not forget to apply operations to both sides of the equation.
When factorising, ensure you have correctly collected all terms containing the subject.
Simplify fractions where possible, but avoid unnecessary expansion if the variable is not inside the bracket.
Mark schemes accept equivalent forms; e.g., (3 − y)/2 is the same as (y − 3)/−2.

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練習題

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1.Make x the subject of 2y = 5x - 7.
Easy
- Ax = (2y + 7)/5
- Bx = (2y - 7)/5
- Cx = (2y + 7)/-5
- Dx = (2y - 7)/-5
2.Rearrange 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
3.Rearrange 2(w + h) = P to make w the subject.
Easy
- Aw = P/2 - h
- Bw = P/2 + h
- Cw = (P - h)/2
- Dw = (P + h)/2
4.Make p the subject of 5p + 7 = m.
Easy
- Ap = (m - 7)/5
- Bp = (m + 7)/5
- Cp = (m - 7)/-5
- Dp = (m + 7)/-5
5.Rearrange 2(4x - y) = 5x - 3 to make y the subject.
Medium
- Ay = (3x + 3)/2
- By = (3x - 3)/2
- Cy = (5x - 3)/2
- Dy = (5x + 3)/2
6.Make t the subject of s = k - t².
Medium
- At = √(k - s)
- Bt = √(s - k)
- Ct = √(k + s)
- Dt = √(k) - s
7.Make y the subject of p = (x + y)/5.
Easy
- Ay = 5p - x
- By = 5p + x
- Cy = (p - x)/5
- Dy = (p + x)/5
8.Make t the subject of 2(d - t) = 4t + 7.
Medium
- At = (2d - 7)/6
- Bt = (2d + 7)/6
- Ct = (2d - 7)/-6
- Dt = (2d + 7)/-6

複習筆記

## Rearranging Formulas ### Subject Appears Once - Use inverse operations (opposite operations) to isolate the variable. - Remove fractions first: multiply both sides by the denominator. - Expand brackets only if the variable is inside them; otherwise, divide by the bracket. - If you get a fraction in a fraction, rewrite as division or multiply top and bottom by the LCD. - Negative denominators: \(\frac{a}{-b} = -\frac{a}{b}\). **Example:** Make \(x\) the subject of \(2y = 5x - 7\). \[5x = 2y + 7 \quad \Rightarrow \quad x = \frac{2y+7}{5}\] ### Subject Appears Twice - Collect all terms with the subject on one side. - Factorise to get the subject once. - If the subject is inside brackets, expand first. - If the subject appears on both sides, bring them together before factorising. - For powers, factorise the power term, then take the appropriate root (remember ± for square roots). **Example:** Make \(x\) the subject of \(k = \frac{2m - x}{x}\). \[kx = 2m - x \quad \Rightarrow \quad kx + x = 2m \quad \Rightarrow \quad x(k+1) = 2m \quad \Rightarrow \quad x = \frac{2m}{k+1}\] ### Key Points - Always perform the same operation on both sides. - Check for multiple correct forms (e.g., \(\frac{y-3}{3}\) is same as \(\frac{y}{3} - 1\)). - For fractions in fractions, simplify carefully. - When square rooting, include \(\pm\).

Subject: The variable that is on its own on one side of the formula. Changing the subject means rearranging to make a different variable the subject.
Inverse operations: Operations that undo each other (e.g., addition and subtraction, multiplication and division). Used to isolate the subject.
Factorising: Writing an expression as a product of its factors. Used when the subject appears more than once to combine terms.
Lowest common denominator (LCD): The smallest multiple shared by denominators. Multiply both sides by the LCD to eliminate fractions.

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