1.6 Exercises

20 points · 4 extra credit

Applications of Linear Systems

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

   $x_1\,\mathrm{Fe} + x_2\,\mathrm{O_2} \longrightarrow x_3\,\mathrm{Fe_2O_3}.$

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

   $\begin{aligned} \text{$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 $x_1$ given that all flows are nonnegative.

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

   $x_1\,\mathrm{C_3H_8} + x_2\,\mathrm{O_2} \longrightarrow x_3\,\mathrm{CO_2} + x_4\,\mathrm{H_2O}.$

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

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

   where each triple is (protein, carbohydrate, fat). Find the numbers of servings $a, b, c$ that provide exactly $7$ g protein, $12$ g carbohydrate, and $5$ g fat.

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

   $x_1\,\mathrm{CO_2} + x_2\,\mathrm{H_2O} \longrightarrow x_3\,\mathrm{C_6H_{12}O_6} + x_4\,\mathrm{O_2}.$

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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.

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

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8. 8. (4 pts ••) Balance the reaction

   $x_1\,\mathrm{KMnO_4} + x_2\,\mathrm{HCl} \longrightarrow x_3\,\mathrm{KCl} + x_4\,\mathrm{MnCl_2} + x_5\,\mathrm{H_2O} + x_6\,\mathrm{Cl_2},$

   finding the smallest positive integer coefficients.

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