Problems — 11.10 Systems of Equations

1

**2×2 system — substitution.** Solve $\begin{cases} y = 2x + 1 \\ 3x + 2y = 16 \end{cases}$ (a) Solve by substitution. (b) Verify your answer by checking both equations.

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2

**2×2 system — elimination.** Solve $\begin{cases} 3x + 2y = 16 \\ 5x - 2y = 8 \end{cases}$ (a) Solve by elimination. (b) Could you have spotted the answer faster? Justify briefly.

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3

**Coffee shop modelling.** A coffee shop sells small drinks for $s$ CHF and large drinks for $\ell$ CHF. - Monday: 30 small + 20 large = CHF 175. - Tuesday: 40 small + 25 large = CHF 230. (a) Write the system. (b) Solve for $s$ and $\ell$. (c) Predict Wednesday revenue: 50 small + 30 large.

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4

**Three-variable system [EXT].** A school orders pens, pencils and rulers. - 5 pens + 3 pencils + 2 rulers = CHF 19 - 2 pens + 4 pencils + 3 rulers = CHF 16 - 1 pen + 2 pencils + 4 rulers = CHF 12 (a) Set up the system. (b) Solve.

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5

**Linear–quadratic system.** Solve $\begin{cases} y = x^2 - 2 \\ y = x + 4 \end{cases}$ (a) Set the equations equal. (b) Solve the resulting quadratic. (c) State the two intersection points.

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6

**System from a graph.** Two lines $L_1$ and $L_2$ are given. $L_1$ has gradient $-2$ and passes through $(0, 5)$. $L_2$ passes through $(0, 1)$ and $(4, 9)$. (a) Find the equations of $L_1$ and $L_2$. (b) Find the intersection point.

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7

**Boat speed.** A boat takes 5 hours to travel 12 km downstream and back upstream. The current is 2 km/h. (a) Let $v$ be the boat's still-water speed. Write the equation modelling the total time. (b) Solve for $v$ to 3 s.f.

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8

**Break-even.** A school trip is offered with two pricing plans. - Plan A: CHF 30 fixed minibus + CHF 8 per student. - Plan B: CHF 50 fixed minibus + CHF 6 per student. (a) Write linear cost equations for each plan. (b) Find the number of students at which both plans cost the same. (c) Which plan is cheaper for 25 students?

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9

**Fit a quadratic [EXT].** A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$, $(2, 9)$, $(4, 21)$. (a) Write three equations. (b) Solve for $a$, $b$, $c$. (c) Predict $y$ at $x = 6$.

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10

**Demand–supply.** In a market, demand $D(p) = -3p + 120$ and supply $S(p) = 2p + 10$, where $p$ is price. (a) Find the equilibrium price and quantity. (b) If a tax of CHF 5 is added per unit (shifting supply up by 5), find the new equilibrium.

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11

**No solution / infinitely many.** Consider $\;2x + 3y = 12, \; 4x + 6y = k$. (a) Find the value of $k$ for which the system has infinitely many solutions. (b) Find the values of $k$ for which the system has no solution. (c) For $k = 30$, sketch both lines and explain what you see.

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12

**Modelling — rates of work [EXT].** A water tank can be filled by pipe A alone in 4 hours, by pipe B alone in 6 hours. A drain D empties the full tank in 8 hours. (a) Express each rate (tank per hour) as a fraction. (b) Set up an equation for $T$, the time to fill the empty tank when all three are open. (c) Solve for $T$ in hours.

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