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The Timpson Lyceum · Linear Algebra (2026)

1.6 Exercises

Problem sheet · 20 points · 4 extra credit

Applications of Linear Systems

  1. 1. (2 pts) Balance the chemical equation by finding the smallest positive integer coefficients:

    x1 Fe+x2 O2⟶x3 Fe2O3.x_1\,\mathrm{Fe} + x_2\,\mathrm{O_2} \longrightarrow x_3\,\mathrm{Fe_2O_3}.

  2. 2. (3 pts) Traffic (cars per hour) flows through four one-way intersections A,B,C,DA, B, C, D. Conservation of flow (in == out) at each intersection gives

    A:x1+40=x2+30B:x2+50=x3+60C:x3+20=x4+40D:x4+30=x1+10.\begin{aligned} \text{$A$:}\quad x_1 + 40 &= x_2 + 30 \\ \text{$B$:}\quad x_2 + 50 &= x_3 + 60 \\ \text{$C$:}\quad x_3 + 20 &= x_4 + 40 \\ \text{$D$:}\quad x_4 + 30 &= x_1 + 10. \end{aligned}

    Solve the system, describe the solution set, and determine the minimum possible value of x1x_1 given that all flows are nonnegative.

  3. 3. (3 pts) Balance the combustion of propane:

    x1 C3H8+x2 O2⟶x3 CO2+x4 H2O.x_1\,\mathrm{C_3H_8} + x_2\,\mathrm{O_2} \longrightarrow x_3\,\mathrm{CO_2} + x_4\,\mathrm{H_2O}.

  4. 4. (3 pts) A nutritionist blends three foods. Per unit serving, the grams of protein, carbohydrate, and fat are:

    Food A:(2,1,0),Food B:(1,3,1),Food C:(0,1,2),\text{Food A}: (2, 1, 0), \quad \text{Food B}: (1, 3, 1), \quad \text{Food C}: (0, 1, 2),

    where each triple is (protein, carbohydrate, fat). Find the numbers of servings a,b,ca, b, c that provide exactly 77 g protein, 1212 g carbohydrate, and 55 g fat.

  5. 5. (3 pts) Balance the photosynthesis reaction:

    x1 CO2+x2 H2O⟶x3 C6H12O6+x4 O2.x_1\,\mathrm{CO_2} + x_2\,\mathrm{H_2O} \longrightarrow x_3\,\mathrm{C_6H_{12}O_6} + x_4\,\mathrm{O_2}.

  6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

    (a) The coefficients that balance a chemical equation are found by solving a homogeneous linear system.

    (b) A network-flow system always has a unique solution.

    (c) In a flow network, the equation at each node states that total flow in equals total flow out.

  7. 7. (3 pts) Explain why the node equations of a closed flow network (like Problem 2) are always dependent — their sum is 0=00 = 0 — so the system has at least one free variable. Interpret what happens to the solution when one branch flow is measured directly.

  8. 8. (4 pts ••) Balance the reaction

    x1 KMnO4+x2 HCl⟶x3 KCl+x4 MnCl2+x5 H2O+x6 Cl2,x_1\,\mathrm{KMnO_4} + x_2\,\mathrm{HCl} \longrightarrow x_3\,\mathrm{KCl} + x_4\,\mathrm{MnCl_2} + x_5\,\mathrm{H_2O} + x_6\,\mathrm{Cl_2},

    finding the smallest positive integer coefficients.