Pack B — Fluency

Mathematics

Year 9 · 9.6 Simultaneous Equations

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

SilverQuestions 11–20

11.
Solve $3x + 2y = 12$ and $x - y = 1$ by elimination.
12.
Solve $y = 4x - 7$ and $2x + y = 5$ .
13.
Solve $2x + y = 7$ and $3x - 2y = 21$ by elimination.
14.
Decide whether the system has unique, no, or infinite solutions: $x + y = 3$ and $2x + 2y = 6$ .
15.
Solve $y = -x + 6$ and $y = 2x - 3$ .
16.
Solve $2(x - 1) + y = 5$ and $3x + 2y = 14$ .
17.
Solve $\dfrac{x}{2} + y = 7$ and $x + 2y = 14$ .
18.
Find the intersection of $3x + 4y = 12$ and $x = 0$ .
19.
Solve $y = 2x + 3$ and $y = -x + 6$ graphically (i.e. find intersection).
20.
Two pens and three pencils cost £2.20. Five pens and one pencil cost £3.10. Find each price.

GoldQuestions 21–30

21.
Use equating $y = y$ method to solve $y = 2x + 1$ and $y = -x + 7$ .
22.
A coach hires a 32-seater bus and a 16-seater bus to take 224 students. Total cost £1900 (32-seater £250, 16-seater £150). Find how many of each.
23.
Concert ticket pricing: Friday 50 adult + 20 child = £790; Saturday 80 adult + 40 child = £1320. Find adult and child prices.
24.
A father is 4× his son's age. In 4 years, he will be 3× his son's age. Find their ages.
25.
Solve $\dfrac{x + y}{3} = 5$ and $\dfrac{x - y}{2} = 1$ .
26.
Solve $2x + y = 10$ and $x^2 + y = 14$ .
27.
A 2-burger meal costs £8.50; a 1-burger + 1-fries meal costs £6.20. Find the prices of a burger and fries.
28.
Three friends share an amount. Alice has £ $x$ , Bob has £ $y$ , Charles £ $z$ . $x + y = 50$ , $y + z = 60$ , $x + z = 70$ . Find each share.
29.
Find the value of $k$ for which $y = 3x + k$ passes through the intersection of $y = 2x + 1$ and $y = x + 4$ .
30.
Find the line through the intersection of $y = 2x + 1$ and $y = -x + 7$ that is parallel to $y = 4x - 5$ .

PlatinumQuestions 31–40

31.
Solve $\dfrac{2x + 3y}{5} = 4$ and $3x - y = 7$ .
32.
Solve $2x + 3y = 19$ and $4x - y = 17$ using (a) substitution, (b) elimination.
33.
Solve $\dfrac{x + 1}{2} - \dfrac{y - 2}{3} = 1$ and $x + y = 5$ .
34.
Two mobile phone tariffs: A: £15 + 10p/min; B: £25 + 5p/min. Find the minutes for equal cost, then which is cheaper for 300 min.
35.
Find the values of $a$ and $b$ such that the system $ax + 2y = 5, 3x + by = 7$ has the solution $(1, 2)$ .
36.
A two-digit number has the digit sum 11 and is 27 greater than the number with reversed digits. Find the original number.
37.
Solve $3x - y = 7$ and $x^2 + y^2 = 25$ .
38.
A 2-variable system has solution $(2, 3)$ . Find the system if both lines pass through this point and one has slope 2 and the other slope $-1$ .
39.
A boat travels 36 km downstream in 2 hours and back upstream in 3 hours. Find the speed of the boat in still water and the speed of the current.
40.
Two mixtures: 60% milk and 40% water in mixture A; 20% milk and 80% water in B. How many litres of each to make 10 L of 50% milk-50% water?