The Timpson Lyceum · Linear Algebra (2026)

1.10 Exercises

Problem sheet · 20 points · 4 extra credit

Linear Models in Business, Science, and Engineering

1. 1. (2 pts) A juice blend is made by mixing $x$ liters of a $20\%$-juice drink with $y$ liters of a $50\%$-juice drink to obtain $9$ liters of a $40\%$-juice drink. Set up and solve the linear system for $x$ and $y$.

2. 2. (3 pts) Two foods A and B are combined to meet nutrient targets. Per serving, Food A supplies $30$ units of protein and $10$ of calcium; Food B supplies $10$ of protein and $40$ of calcium. Find the numbers of servings $a$ and $b$ that provide exactly $90$ units of protein and $140$ units of calcium.

3. 3. (3 pts) Each year $10\%$ of a city's population moves to its suburbs and $20\%$ of the suburban population moves to the city (no other change). Writing the population as $\mathbf{x} = \begin{bmatrix} \text{city} \\ \text{suburb} \end{bmatrix}$ (in thousands), the one-year update is $\mathbf{x}_{1} = M\mathbf{x}_0$ with

   $M = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix}.$

   If $\mathbf{x}_0 = \begin{bmatrix} 600 \\ 400 \end{bmatrix}$, compute $\mathbf{x}_1$ and verify the total population is unchanged.

4. 4. (3 pts) In a circuit, three currents satisfy Kirchhoff's laws:

   $\begin{aligned} I_1 - I_2 - I_3 &= 0 \\ 2I_1 + 3I_2 &= 12 \\ 3I_2 + 4I_3 &= 10. \end{aligned}$

   Find $I_1, I_2, I_3$.

5. 5. (3 pts) An investor splits $10,000 among three funds A, B, C with annual returns $5\%$, $7\%$, $10\%$. She wants a total return of $680 and decides to place twice as much in A as in C. Letting $x, y, z$ be the amounts in A, B, C, set up and solve the system.

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

   (a) In a migration (stochastic) matrix like $M$ above, each column sums to $1$.

   (b) Every linear model has a unique solution.

   (c) In a diet model, each equation corresponds to one nutrient constraint.

7. 7. (3 pts) Explain what each column of the migration matrix $M = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix}$ represents, and explain why each column must sum to $1$.

8. 8. (4 pts ••) Find the population distribution, expressed as fractions of the total, that is unchanged from one year to the next under $M$ above — i.e. the vector $\mathbf{x}$ with $M\mathbf{x} = \mathbf{x}$ and entries summing to $1$. Verify that $M\mathbf{x} = \mathbf{x}$ for your answer.