Mathematics
EN FR
← Back to review

Problem-solving

11.10 Systems of Equations

Show all working. Partial marks are given for method.

  1. 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.

    Working space

  2. 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.

    Working space

  3. 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.

    Working space

  4. 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.

    Working space

  5. 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.

    Working space

  6. 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.

    Working space

  7. 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.

    Working space

  8. 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?

    Working space

  9. 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 $x = 6$.

    Working space

  10. 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.

    Working space

  11. 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.

    Working space

  12. 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.

    Working space