9.6 Simultaneous Equations

Show all working. Partial marks are given for method.

1

**Solve simultaneously.** Solve the system $y = 2x + 1$ and $3x + y = 11$, and verify by substitution into both equations.

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2

**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
3

**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
4

**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
5

**Three methods.** Solve $2x + 3y = 19$ and $4x - y = 17$ using: (a) Substitution (b) Elimination (c) Equating $y = y$ (or graphical)

Working space
6

**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
7

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

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8

**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
9

**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
10

**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.

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11

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

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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.

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