Problem-solving Pack

Mathematics

Year 9 · 9.6 Simultaneous Equations

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

Solve simultaneously. Solve the system

y = 2x + 1

and

3x + y = 11

, and verify by substitution into both equations.

Working space

2Problem 2 of 12

Man and son. A man is currently 4× his son's age. In 4 years, he will be 3× as old.

(a) Let son's age be

s

. Write an equation.
(b) Solve to find both ages.

Working space

3Problem 3 of 12

Concert ticket pricing. Friday: 50 adult + 20 child = £790. Saturday: 80 adult + 40 child = £1320.

(a) Write 2 simultaneous equations.
(b) Solve them.

Working space

4Problem 4 of 12

Mobile phone tariffs. Tariff A: £15 + 10p/min. Tariff B: £25 + 5p/min.

(a) Write equations for total cost

C

in terms of minutes

m

.
(b) Find

m

for equal cost.
(c) Which is cheaper for 300 min/month?

Working space

5Problem 5 of 12

Three methods. Solve

2x + 3y = 19

and

4x - y = 17

using:

(a) Substitution
(b) Elimination
(c) Equating

y = y

(or graphical)

Working space

6Problem 6 of 12

Boat and current. A boat travels 36 km downstream in 2 hours and back upstream in 3 hours.

(a) Find the speeds (downstream and upstream).
(b) Set up simultaneous equations for boat speed

b

and current

c

.
(c) Solve.

Working space

7Problem 7 of 12

Mixture problem. A 10 L solution is 30% acid. How much pure water should be added to make it 20% acid?

Working space

8Problem 8 of 12

Sum and product. Two numbers have sum 14 and product 48.

(a) Set up simultaneous equations.
(b) Solve.
(c) The numbers are the roots of a quadratic. State it.

Working space

9Problem 9 of 12

No solution or infinite solutions. Consider the systems:

(a)

x + y = 5

and

2x + 2y = 12

.
(b)

x + y = 5

and

2x + 2y = 10

.

For each, decide whether it has a unique solution, no solution, or infinite. Justify.

Working space

10Problem 10 of 12

Two-digit reversal. A two-digit number has digit sum 11. When the digits are reversed, the new number is 27 less than the original.

(a) Set up two equations in

a

(tens) and

b

(units).
(b) Solve.

Working space

11Problem 11 of 12

Geometry application. A rectangle has perimeter 30 cm and area 50 cm². Find its dimensions.

Working space

12Problem 12 of 12

Three friends sharing. Alice, Bob, Charles share a sum.

A + B = 50, B + C = 60, A + C = 70

.

(a) Add all three equations and find

A + B + C

.
(b) Find each share.

Working space