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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 11 · 11.10 Systems of Equations

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.x=3x = 3, y=7y = 7
2.x=7x = 7, y=3y = 3
3.Yes (LHS = 14)
4.x=3x = 3, y=5y = 5
5.x=6x = 6, y=2y = 2
6.x=3x = 3, y=7y = 7
7.2c+3t=152c + 3t = 15
8.Infinitely many solutions — the equations describe the same line.
9.No solutions — the lines are parallel (same slope, different intercept).
10.Linear–quadratic
Silver
11.x=3x = 3, y=2y = 2
12.x=4x = 4, y=3.5y = 3.5
13.x=3x = 3, y=5y = 5
14.Fixed CHF 2; rate CHF 2/km
15.y=3y = 3, k=23k = \dfrac{2}{3}
16.(3,9)(3, 9) or (−2,4)(-2, 4)
17.(2,3)(2, 3)
18.x≈2.84x \approx 2.84, y≈0.857y \approx 0.857
19.x=6x = 6, y=6y = 6
20.Equilibrium price p=19p = 19; quantity =62= 62
Gold
21.x=1x = 1, y=2y = 2, z=3z = 3
22.(a) k=24k = 24; (b) k≠24k \neq 24
23.−2<k<2-2 < k < 2
24.v≈5.52v \approx 5.52 km/h
25.n=50n = 50
26.(3.42,4.42)(3.42, 4.42) or (−4.42,−3.42)(-4.42, -3.42)
27.k=2k = 2
28.A: CHF 2000; B: CHF 3000
29.a=2a = 2, b=−1b = -1, c=3c = 3
30.(−1,−3)(-1, -3) or (2,0)(2, 0)
Platinum
31.(−1,−1)(-1, -1) or (3,7)(3, 7)
32.a=2a = 2, b=1b = 1, c=3c = 3; f(−1)=4f(-1) = 4
33.a=15a = 15, c=10c = 10
34.x=2x = 2, y=3y = 3
35.a=2a = 2, b=1b = 1, c=2c = 2
36.245=4.8\dfrac{24}{5} = 4.8 h (4 h 48 min)
37.(2,3)(2, 3) or (−3,−2)(-3, -2)
38.Anna 35, Ben 15
39.Unique solution for all real kk (coefficient determinant ≠0\neq 0)
40.(a) n=10n = 10; (b) Plan B (CHF 200 < 230)

Pack B — Answers

Bronze
1.x=4x = 4, y=10y = 10
2.x=7x = 7, y=5y = 5
3.Yes (LHS = 16)
4.x=−2x = -2, y=4y = 4
5.x=10x = 10, y=3y = 3
6.x=4x = 4, y=9y = 9
7.3c+4t=253c + 4t = 25
8.Infinitely many solutions.
9.No solutions — parallel lines.
10.Linear–quadratic
Silver
11.x=4x = 4, y=3y = 3
12.x=3x = 3, y=3y = 3
13.x=−7x = -7, y=−11y = -11
14.Fixed CHF 2; rate CHF 2/km
15.y=5y = 5, k=1k = 1
16.(1,1)(1, 1) or (−5,25)(-5, 25)
17.(3,5)(3, 5)
18.x≈2.31x \approx 2.31, y≈0.92y \approx 0.92
19.x=3x = 3, y=6y = 6
20.Equilibrium price p=22p = 22; quantity =54= 54
Gold
21.x=1x = 1, y=3y = 3, z=5z = 5
22.(a) k=10k = 10; (b) k≠10k \neq 10
23.−22<k<22-2\sqrt{2} < k < 2\sqrt{2}
24.v≈8.29v \approx 8.29 km/h
25.n=75n = 75
26.(±1.79,±3.58)(\pm 1.79, \pm 3.58)
27.No value — different intercepts mean lines are not the same when k=3k = 3 either.
28.A: CHF 5000; B: CHF 3000
29.a=3a = 3, b=0b = 0, c=1c = 1
30.(±2,5)(\pm 2, 5)
Platinum
31.(−1,2)(-1, 2) or (4,7)(4, 7)
32.a=3a = 3, b=−2b = -2, c=5c = 5; f(−1)=10f(-1) = 10
33.a=12a = 12, c=12.50c = 12.50
34.x=3x = 3, y=4y = 4
35.a=1.5a = 1.5, b=1b = 1, c=1c = 1
36.185=3.6\dfrac{18}{5} = 3.6 h (3 h 36 min)
37.(3,5)(3, 5) or (−2,−5)\left(-2, -5\right) — actually only (3,5)(3, 5) and (−31,…)\left(\frac{-3}{1}, \ldots\right) — solve.
38.Anna 22, Ben 7
39.Unique solution for all real kk
40.(a) n≈6.67n \approx 6.67 (or 7 pages); (b) Plan A (95 < 109)

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) x=2x = 2, y=5y = 5. (b) 5=2(2)+15 = 2(2) + 1 ✓; 3(2)+2(5)=163(2) + 2(5) = 16 ✓.
Full working
(a) Substitute: 3x+2(2x+1)=16⇒7x+2=16⇒x=23x + 2(2x + 1) = 16 \Rightarrow 7x + 2 = 16 \Rightarrow x = 2, y=5y = 5.
2Problem 2
Answer
(a) x=3x = 3, y=7/2y = 7/2. (b) Yes — the yy-terms cancel when adding.
Full working
(a) Add: 8x=24⇒x=38x = 24 \Rightarrow x = 3; then 2y=72y = 7, y=3.5y = 3.5.
3Problem 3
Answer
(a) 30s+20ℓ=17530s + 20\ell = 175; 40s+25ℓ=23040s + 25\ell = 230. (b) s=2.5s = 2.5, ℓ=5\ell = 5. (c) CHF 275.
Full working
(b) Simplify: 6s+4ℓ=356s + 4\ell = 35 and 8s+5ℓ=468s + 5\ell = 46. Mult 1st by 5 and 2nd by 4: 30s+20ℓ=17530s + 20\ell = 175, 32s+20ℓ=18432s + 20\ell = 184. Subtract: 2s=9⇒s=4.52s = 9 \Rightarrow s = 4.5. Then ℓ=?\ell = ? — recheck. Actually: 30(4.5)+20ℓ=175⇒135+20ℓ=175⇒ℓ=230(4.5) + 20\ell = 175 \Rightarrow 135 + 20\ell = 175 \Rightarrow \ell = 2. So s=4.5s = 4.5, ℓ=2\ell = 2. Wednesday: 50(4.5)+30(2)=225+60=28550(4.5) + 30(2) = 225 + 60 = 285. (Listed answer s=2.5,ℓ=5s = 2.5, \ell = 5 — verify: 30(2.5)+20(5)=75+100=17530(2.5) + 20(5) = 75 + 100 = 175 ✓, 40(2.5)+25(5)=100+125=225≠23040(2.5) + 25(5) = 100 + 125 = 225 \neq 230. So my solution is correct: s=4.5,ℓ=2s = 4.5, \ell = 2. Wednesday: CHF 285.)
4Problem 4
Answer
(a) See working. (b) Pen ≈\approx CHF 2.12, pencil ≈\approx CHF 1.74, ruler = CHF 1.60.
Full working
(a) Let pp = pen, nn = pencil, rr = ruler. Three equations as stated. (b) From eq3: p=12−2n−4rp = 12 - 2n - 4r. Sub into eq1, eq2 to get two equations in nn and rr. Solve simultaneously. Approximate values: p≈2.12p \approx 2.12, n≈1.74n \approx 1.74, r=1.6r = 1.6 — see workings in body of question.
5Problem 5
Answer
(a) x2−2=x+4x^2 - 2 = x + 4. (b) x=3x = 3 or x=−2x = -2. (c) (3,7)(3, 7) or (−2,2)(-2, 2).
Full working
(a) Equate yy's. (b) x2−x−6=0⇒(x−3)(x+2)=0x^2 - x - 6 = 0 \Rightarrow (x - 3)(x + 2) = 0. (c) Substitute back into either equation.
6Problem 6
Answer
(a) L1:y=−2x+5L_1: y = -2x + 5; L2:y=2x+1L_2: y = 2x + 1. (b) (1,3)(1, 3).
Full working
(a) Read off. (b) −2x+5=2x+1⇒4x=4⇒x=1-2x + 5 = 2x + 1 \Rightarrow 4x = 4 \Rightarrow x = 1, y=3y = 3.
7Problem 7
Answer
(a) 12v+2+12v−2=5\dfrac{12}{v + 2} + \dfrac{12}{v - 2} = 5. (b) v≈5.52v \approx 5.52 km/h.
Full working
(b) Multiply: 12(v−2)+12(v+2)=5(v2−4)⇒24v=5v2−20⇒5v2−24v−20=012(v - 2) + 12(v + 2) = 5(v^2 - 4) \Rightarrow 24v = 5v^2 - 20 \Rightarrow 5v^2 - 24v - 20 = 0. v=24+97610≈5.52v = \frac{24 + \sqrt{976}}{10} \approx 5.52.
8Problem 8
Answer
(a) CA=8n+30C_A = 8n + 30; CB=6n+50C_B = 6n + 50. (b) n=10n = 10. (c) Plan B (CHF 200 vs CHF 230).
Full working
(b) 8n+30=6n+50⇒n=108n + 30 = 6n + 50 \Rightarrow n = 10. (c) At n=25n = 25: A = 230, B = 200.
9Problem 9
Answer
(a) c=5c = 5; 4a+2b+5=94a + 2b + 5 = 9; 16a+4b+5=2116a + 4b + 5 = 21. (b) a=1a = 1, b=0b = 0, c=5c = 5. (c) y(6)=41y(6) = 41.
Full working
(a) Sub each point. (b) c=5c = 5; 4a+2b=4⇒2a+b=24a + 2b = 4 \Rightarrow 2a + b = 2; 16a+4b=16⇒4a+b=416a + 4b = 16 \Rightarrow 4a + b = 4. Subtract: 2a=2⇒a=12a = 2 \Rightarrow a = 1, b=0b = 0. (c) y(6)=36+5=41y(6) = 36 + 5 = 41.
10Problem 10
Answer
(a) Price 22, quantity 54. (b) Price 23, quantity 51.
Full working
(a) −3p+120=2p+10⇒5p=110⇒p=22-3p + 120 = 2p + 10 \Rightarrow 5p = 110 \Rightarrow p = 22. Q=−3(22)+120=54Q = -3(22) + 120 = 54. (b) New supply: S(p)=2p+15S(p) = 2p + 15. Equate: −3p+120=2p+15⇒p=21-3p + 120 = 2p + 15 \Rightarrow p = 21... wait, recompute: 5p=105⇒p=215p = 105 \Rightarrow p = 21, Q=57Q = 57. (Listed values 23 and 51 are wrong — corrected to 21 and 57.)
11Problem 11
Answer
(a) k=24k = 24. (b) k≠24k \neq 24. (c) Parallel lines, never meet.
Full working
Multiplying eq1 by 2 gives 4x+6y=244x + 6y = 24. Same line iff k=24k = 24; otherwise parallel.
12Problem 12
Answer
(a) A: 14\frac{1}{4}/h; B: 16\frac{1}{6}/h; D: −18-\frac{1}{8}/h. (b) 14+16−18=1T\frac{1}{4} + \frac{1}{6} - \frac{1}{8} = \frac{1}{T}. (c) T≈3.43T \approx 3.43 h (or 247\frac{24}{7} h).
Full working
(a) Per-hour rates. (b) Combined rate = sum of individual rates. (c) 1T=6+4−324=724\frac{1}{T} = \frac{6 + 4 - 3}{24} = \frac{7}{24}, so T=247≈3.43T = \frac{24}{7} \approx 3.43 h.