Changing the Subject: Advanced
Formula rearrangement — 31 items
Prepared by Miss Clarissa Ng
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www.clartutors.com
Attempt each part first — the working space is there for your own line of reasoning. The full worked solutions follow at the end, one line per step, so you can compare your method at every stage rather than only the final answer.
Q1(a) A = ½(x + y)
Q1(b) A = πr² + 2πrx
Q1(c) T = 3x² − y
Q1(d) s = max
Q1(e) s = uy + ½xy²
Q1(f) ⅓w = ¼x + t
Q1(g) j = x + 3d
Q1(h) g = tx − 2
Q1(i) p = 3(y + 2x)²
Q1(j) 12w = ¾(2x + a)
Q2(a) 5(m + y) = 4(m − 3y)
Q2(b) 3(3m + 4) = 7(m + 2a)
Q2(c) 15(2m + 3c) = 5(m + 7c)
Q2(d) 9m + 4c = 2(a + 3m)
Q2(e) a(c + m) = 2(c + 3m)
Q2(f) w(m + n) = x(m − n)
Q2(g) 8 = m + 3cm − f
Q2(h) y = m + 4m + 5
Q2(i) y = 3mt − a²m
Q2(j) r(c + 7) = 3m + 5
Q2(k) x = 4πm + am
Q2(l) 2 = m + km − t
Q2(m) dm = y − em
Q2(n) m(c + d) = m + f
Q2(o) y − mp = np + 2y
Q2(p) m(r + p) = r(h − m)
Q2(q) πx = m + 8m − 1
Q2(r) 3m + 2c = m + 1a
Q3(a) w = aca − c
Q3(b) w = 6 + ac + 2
Apply 1 The cosine rule is a² = b² + c² − 2bc cos A. Make cos A the subject.
Every line shows the operation and what it gives, ending in the finished formula. The rearrangements were re-checked by solving each original equation for its subject.
| Q1(a) | ×2: 2A = x + y → subtract y x = 2A − y |
| Q1(b) | subtract πr²: A − πr² = 2πrx → divide by 2πr x = A − πr²2πr |
| Q1(c) | add y: T + y = 3x² → divide by 3: T + y3 = x² → square root x = √T + y3 |
| Q1(d) | ×ax: asx = m → divide by as x = mas |
| Q1(e) | subtract uy: s − uy = ½xy² → ×2: 2s − 2uy = xy² → divide by y² x = 2s − 2uyy² |
| Q1(f) | subtract t: ⅓w − t = ¼x → ×4 x = 43w − 4t |
| Q1(g) | ×d: dj = x + 3 → subtract 3 x = dj − 3 |
| Q1(h) | ×(x − 2): g(x − 2) = t → divide by g: x − 2 = tg → add 2 x = tg + 2 |
| Q1(i) | divide by 3: p3 = (y + 2x)² → square root: √p3 = y + 2x → subtract y → divide by 2 x = (√p3 − y) ÷ 2 |
| Q1(j) | ×4/3: 16w = 2x + a → subtract a: 16w − a = 2x → divide by 2 x = 8w − ½a |
| Q2(a) | expand: 5m + 5y = 4m − 12y → 5m − 4m = −12y − 5y m = −17y |
| Q2(b) | expand: 9m + 12 = 7m + 14a → 9m − 7m = 14a − 12 → divide by 2 m = 7a − 6 |
| Q2(c) | expand: 30m + 45c = 5m + 35c → 25m = −10c → divide by 25 m = −25c |
| Q2(d) | expand: 9m + 4c = 2a + 6m → 3m = 2a − 4c → divide by 3 m = 23(a − 2c) |
| Q2(e) | expand: ac + am = 2c + 6m → am − 6m = 2c − ac → m(a − 6) = c(2 − a) m = c(2 − a)a − 6 |
| Q2(f) | expand: wm + wn = xm − xn → wm − xm = −xn − wn → m(w − x) = −n(x + w) → divide by (w − x) m = n(w + x)x − w |
| Q2(g) | ×(m − f): 8m − 8f = m + 3c → 8m − m = 3c + 8f → divide by 7 m = 3c + 8f7 |
| Q2(h) | ×(m + 5): ym + 5y = m + 4 → ym − m = 4 − 5y → m(y − 1) = 4 − 5y m = 4 − 5yy − 1 |
| Q2(i) | factorise: y = m(3t − a²) → divide by (3t − a²) m = y3t − a² |
| Q2(j) | expand: rc + 7r = 3m + 5 → rc + 7r − 5 = 3m → divide by 3 m = cr + 7r − 53 |
| Q2(k) | factorise: x = m(4π + a) → divide by (a + 4π) m = xa + 4π |
| Q2(l) | ×(m − t): 2m − 2t = m + k → 2m − m = k + 2t m = k + 2t |
| Q2(m) | add em: dm + em = y → m(d + e) = y → divide by (d + e) m = yd + e |
| Q2(n) | expand: mc + md = m + f → mc + md − m = f → m(c + d − 1) = f m = fc + d − 1 |
| Q2(o) | subtract y: −mp = np + y → −mp − np = y → −p(m + n) = y → divide by −p m = −n − yp |
| Q2(p) | expand: mr + mp = rh − rm → mr + rm = rh − mp → 2rm + mp = rh → m(2r + p) = rh m = hrp + 2r |
| Q2(q) | ×(m − 1): πxm − πx = m + 8 → m(πx − 1) = πx + 8 m = πx + 8πx − 1 |
| Q2(r) | cross-multiply: a(3m + 2) = c(m + 1) → 3am + 2a = cm + c → 3am − cm = c − 2a → m(3a − c) = c − 2a m = c − 2a3a − c |
| Q3(a) | ×(a − c): w(a − c) = ac → wa − wc = ac → wa = ac + wc = c(a + w) c = awa + w |
| Q3(b) | subtract 6: w − 6 = ac + 2 → ×(c + 2): (w − 6)(c + 2) = a → divide by (w − 6) → subtract 2 c = a − 2w + 12w − 6 |
| Apply 1 | add 2bc cos A: a² + 2bc cos A = b² + c² → subtract a²: 2bc cos A = b² + c² − a² → divide by 2bc cos A = b² + c² − a²2bc |