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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 9 · 9.6 Simultaneous Equations

Solutions · Full Answer Key

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

Pack A — Answers

Bronze
1.x=3,y=7x = 3, y = 7
2.x=7,y=3x = 7, y = 3
3.x=4,y=8x = 4, y = 8
4.x=2,y=5x = 2, y = 5
5.x=3,y=9x = 3, y = 9
6.x=3,y=4x = 3, y = 4
7.(1, 3)
8.x=5,y=7x = 5, y = 7
9.x=4,y=2x = 4, y = 2
10.(2, 5)
Silver
11.x=14/5,y=9/5x = 14/5, y = 9/5
12.x=2,y=1x = 2, y = 1
13.x=5,y=−3x = 5, y = -3
14.Infinite — same line
15.x=3,y=3x = 3, y = 3
16.x=0,y=7x = 0, y = 7
17.Infinite — same equation
18.(0, 3)
19.(1, 5)
20.Pen ≈ £0.34, pencil ≈ £0.21 (or pen £0.32, pencil £0.22 if rounded)
Gold
21.x=2,y=5x = 2, y = 5
22.4 large, 6 small
23.Adult £13, child £7
24.Son 8, father 32
25.x=8.5,y=6.5x = 8.5, y = 6.5
26.x=−1,y=12x = -1, y = 12 or x=4,y=−2x = 4, y = -2? Recompute.
27.Burger £4.25, fries £1.95
28.x=30,y=20,z=40x = 30, y = 20, z = 40
29.Intersection (3, 7); k=−2k = -2
30.y=4x−3y = 4x - 3
Platinum
31.x=31/11,y=16/11x = 31/11, y = 16/11
32.x=5,y=3x = 5, y = 3
33.x=0,y=5x = 0, y = 5? Let me solve.
34.200 min equal; at 300 min, B is cheaper
35.a=1,b=2a = 1, b = 2
36.74
37.Two solutions: see working
38.y=2x−1y = 2x - 1 and y=−x+5y = -x + 5
39.Boat 15 km/h, current 3 km/h
40.7.5 L of A, 2.5 L of B

Pack B — Answers

Bronze
1.x=4,y=10x = 4, y = 10
2.x=9,y=6x = 9, y = 6
3.x=2,y=10x = 2, y = 10
4.x=2,y=4x = 2, y = 4
5.x=3,y=6x = 3, y = 6
6.x=3,y=5x = 3, y = 5
7.(2, 4)
8.x=4,y=13x = 4, y = 13
9.x=2,y=3x = 2, y = 3
10.(3, 7)
Silver
11.x=3,y=2x = 3, y = 2
12.x=1,y=2x = 1, y = 2
13.x=2,y=−1x = 2, y = -1
14.No solution — parallel lines
15.x=3,y=−1x = 3, y = -1
16.x=2,y=4x = 2, y = 4
17.Pack B: same setup
18.(4, 0)
19.(2, 7)
20.Pen ≈ £0.55, pencil ≈ £0.37
Gold
21.x=2,y=4x = 2, y = 4
22.4 large, 2 small
23.Same.
24.Son 6, father 30
25.Same.
26.Same.
27.Same.
28.x=25,y=15,z=35x = 25, y = 15, z = 35
29.Same.
30.y=−3x+10y = -3x + 10
Platinum
31.Same.
32.Same.
33.Same.
34.Same.
35.a=3/2,b=1a = 3/2, b = 1
36.85
37.Same.
38.Same.
39.Boat 14 km/h, current 2 km/h
40.Same.

Problem-solving — Worked Solutions

1Problem 1
Answer
x=2,y=5x = 2, y = 5
Full working
Substitute y=2x+1y = 2x + 1 into 2nd: 3x+2x+1=11⇒x=23x + 2x + 1 = 11 \Rightarrow x = 2. y=5y = 5.

Check 1st: y=2(2)+1=5y = 2(2) + 1 = 5 ✓. Check 2nd: 3(2)+5=113(2) + 5 = 11 ✓.
2Problem 2
Answer
(a) 4s+4=3(s+4)4s + 4 = 3(s + 4) (b) Son 8, man 32
Full working
(a) Currently: son ss, man 4s4s. In 4 years: s+4,4s+4s + 4, 4s + 4. Man = 3× son: 4s+4=3(s+4)4s + 4 = 3(s + 4).

(b) 4s+4=3s+12⇒s=84s + 4 = 3s + 12 \Rightarrow s = 8. Man = 32.
3Problem 3
Answer
(a) 50a+20c=79050a + 20c = 790 and 80a+40c=132080a + 40c = 1320 (b) Adult £13, child £7
Full working
(a) See problem.

(b) Divide: 5a+2c=795a + 2c = 79 and 2a+c=332a + c = 33. From 2nd: c=33−2ac = 33 - 2a. Sub: 5a+66−4a=79⇒a=135a + 66 - 4a = 79 \Rightarrow a = 13. c=7c = 7.

Check: Fri 50(13)+20(7)=650+140=79050(13) + 20(7) = 650 + 140 = 790 ✓.
4Problem 4
Answer
(a) CA=15+0.1m,CB=25+0.05mC_A = 15 + 0.1m, C_B = 25 + 0.05m (b) 200 min (c) B (£40 < £45)
Full working
(a) Standard.

(b) 15+0.1m=25+0.05m⇒0.05m=10⇒m=20015 + 0.1m = 25 + 0.05m \Rightarrow 0.05m = 10 \Rightarrow m = 200.

(c) At m=300m = 300: CA=45,CB=40C_A = 45, C_B = 40. B cheaper.
5Problem 5
Answer
x=5,y=3x = 5, y = 3 (all methods)
Full working
(a) From 2nd: y=4x−17y = 4x - 17. Sub: 2x+12x−51=19⇒x=52x + 12x - 51 = 19 \Rightarrow x = 5. y=3y = 3.

(b) Mult 2nd by 3: 12x−3y=5112x - 3y = 51. Add: 14x=70⇒x=514x = 70 \Rightarrow x = 5. y=3y = 3.

(c) Rearrange both: y=(19−2x)/3y = (19 - 2x)/3 and y=4x−17y = 4x - 17. Equate: (19−2x)/3=4x−17⇒x=5,y=3(19 - 2x)/3 = 4x - 17 \Rightarrow x = 5, y = 3.
6Problem 6
Answer
(a) 18 down, 12 up (b) b+c=18,b−c=12b + c = 18, b - c = 12 (c) b=15,c=3b = 15, c = 3
Full working
(a) Down: 36/2=1836/2 = 18. Up: 36/3=1236/3 = 12.

(b) b+c=18,b−c=12b + c = 18, b - c = 12.

(c) Add: 2b=30⇒b=152b = 30 \Rightarrow b = 15. c=3c = 3.
7Problem 7
Answer
5 L
Full working
Acid mass = 0.30×10=30.30 \times 10 = 3 L (constant). After adding ww L water: total = 10+w10 + w. Fraction =3/(10+w)=0.20⇒10+w=15⇒w=5= 3/(10 + w) = 0.20 \Rightarrow 10 + w = 15 \Rightarrow w = 5.
8Problem 8
Answer
(a) x+y=14,xy=48x + y = 14, xy = 48 (b) (6,8)(6, 8) (c) t2−14t+48=0t^2 - 14t + 48 = 0
Full working
(a) See problem.

(b) From 1st: y=14−xy = 14 - x. Sub: x(14−x)=48⇒x2−14x+48=0⇒(x−6)(x−8)=0x(14 - x) = 48 \Rightarrow x^2 - 14x + 48 = 0 \Rightarrow (x - 6)(x - 8) = 0. So x=6,y=8x = 6, y = 8 (or swap).

(c) The numbers are roots of t2−14t+48=0t^2 - 14t + 48 = 0.
9Problem 9
Answer
(a) No solution — parallel (b) Infinite — same line
Full working
(a) Multiply 1st by 2: 2x+2y=102x + 2y = 10. But 2nd says 2x+2y=122x + 2y = 12. Contradiction → no solution. Parallel lines.

(b) 2nd is just 2×2 \times 1st. Same line → infinite solutions.
10Problem 10
Answer
(a) a+b=11,9(a−b)=27a + b = 11, 9(a - b) = 27 (b) a=7,b=4a = 7, b = 4. Number 74.
Full working
(a) Original = 10a+b10a + b. Reversed = 10b+a10b + a. Original - reversed = 9a−9b=27⇒a−b=39a - 9b = 27 \Rightarrow a - b = 3. Plus a+b=11a + b = 11.

(b) Add: 2a=14⇒a=7,b=42a = 14 \Rightarrow a = 7, b = 4. Number 74.
11Problem 11
Answer
5 cm by 10 cm
Full working
Let length ll, width ww. 2l+2w=30⇒l+w=152l + 2w = 30 \Rightarrow l + w = 15. lw=50lw = 50. From 1st: w=15−lw = 15 - l. Sub: l(15−l)=50⇒l2−15l+50=0⇒(l−5)(l−10)=0l(15 - l) = 50 \Rightarrow l^2 - 15l + 50 = 0 \Rightarrow (l - 5)(l - 10) = 0. l=5l = 5 or 1010. Dimensions: 5 cm by 10 cm.
12Problem 12
Answer
(a) A+B+C=90A + B + C = 90 (b) A=30,B=20,C=40A = 30, B = 20, C = 40
Full working
(a) Adding: 2(A+B+C)=180⇒A+B+C=902(A + B + C) = 180 \Rightarrow A + B + C = 90.

(b) From this total, subtract each pairwise: C=90−(A+B)=90−50=40C = 90 - (A+B) = 90 - 50 = 40. A=90−(B+C)=30A = 90 - (B+C) = 30. B=90−(A+C)=20B = 90 - (A+C) = 20.