Solving Problems using Linear Equations and Mathematical Modeling

Mathematical Model of a Problem

• Definition of Mathematical Modeling: To solve a practical problem, it is recommended to first construct its mathematical model. A mathematical model represents the relationships between known and unknown quantities using mathematical concepts, relations, formulas, equations, and expressions.

Algorithm for Solving Word Problems Using Linear Equations

• Sequential Procedure: To solve a word problem by constructing a linear equation, follow these six consecutive steps:
  • Step 1: Designate one of the unknown quantities using a variable (e.g., $x$).
  • Step 2: Express any other unknown quantities in terms of the introduced variable.
  • Step 3: Establish the mathematical relationship between the known values and unknown quantities based on the problem conditions, and construct the corresponding equation.
  • Step 4: Solve the constructed equation.
  • Step 5: Analyze the solution(s) of the equation to determine the primary unknown quantity, and calculate any additional unknown values as required by the problem.
  • Step 6: State and write down the final answer to the problem explicitly.

Fully Worked-Out Sample Problems

• Problem 1: Twin Sisters' SMS Messages

  • Condition: On their birthday, twin sisters Natalya and Olena received a total of 127 greeting SMS messages combined. Natalya received 13 more messages than Olena. Determine how many SMS messages each sister received.
  • Mathematical Modeling & Solution:
  • Let $x$ represent the number of SMS messages received by Olena.
  • Express the number of messages received by Natalya in terms of $x$:       $(x + 13)$
  • Express the total combined number of messages received by both sisters:       $(x + x + 13)$
  • Equate the total combined expression to the given total of 127 messages to form the equation:       $x + x + 13 = 127$
  • Simplify and solve the equation step-by-step:       $2x + 13 = 127$$2x = 114$$x = 57$
  • Calculate Natalya's total messages:       $57 + 13 = 70$
  • Conclusion & Answer: Olena received 57 messages and Natalya received 70 messages. Answer: 70 messages; 57 messages.

• Problem 2: Calculation of Minimum Salary for Bank Loan Approval

  • Condition: A bank calculates the maximum allowable loan amount $S$ using the formula:     S = \n\n\frac{C}{3} \times n     where $S$ is the total loan amount in UAH, $C$ is the borrower's average monthly salary in UAH, and $n$ is a coefficient determined by the loan duration:
  • For a 1-year loan term: $n = 9$
  • For a 2-year loan term: $n = 21$
  • For a 3-year loan term: $n = 33$     Find the lowest average monthly salary $C$ (denoted as $x\,\text{UAH}$) required for a borrower to qualify for a loan of $30\,000\,\text{UAH}$ across all three terms.
  • Case 1: Loan Term of 1 Year ($n = 9$):
  • Set up the equation using $S = 30\,000\,\text{UAH}$:       $30\,000 = \frac{x}{3} \times 9$
  • Simplify the right side:       $30\,000 = 3x$
  • Solve for $x$:       $x = 10\,000$
  • Conclusion: The borrower's average monthly salary must be at least $10\,000\,\text{UAH}$.
  • Case 2: Loan Term of 2 Years ($n = 21$):
  • Set up the equation:       $30\,000 = \frac{x}{3} \times 21$
  • Simplify the right side:       $30\,000 = 7x$
  • Solve for $x$:       $x \approx 4285.7$
  • Conclusion: Rounding up to ensure minimum qualification, the average monthly salary must be at least $4286\,\text{UAH}$.
  • Case 3: Loan Term of 3 Years ($n = 33$):
  • Set up the equation:       $30\,000 = \frac{x}{3} \times 33$
  • Simplify the right side:       $30\,000 = 11x$
  • Solve for $x$:       $x \approx 2727.3$
  • Conclusion: Rounding up to ensure minimum qualification, the average monthly salary must be at least $2728\,\text{UAH}$.
  • Answer Summary:     1) $10\,000\,\text{UAH}$     2) $4286\,\text{UAH}$     3) $2728\,\text{UAH}$

• Problem 3: Opposite-Direction Motion of Vehicles

  • Condition: A truck departed from Town A heading towards Town B, which are separated by a distance of $310\,\text{km}$. Exactly 30 minutes ($0.5\,\text{h}$) later, a passenger car departed from Town B heading towards Town A. The two vehicles met 2 hours after the passenger car departed. Determine the speed of each vehicle, given that the passenger car's speed is $20\,\text{km/h}$ greater than the truck's speed.
  • Tabular Breakdown of Motion Parameters:
  • Let $x\,\text{km/h}$ be the speed of the truck.
  • Speed of passenger car: $(x + 20)\,\text{km/h}$.
  • Time traveled by truck: $2\,\text{h} + 0.5\,\text{h} = 2.5\,\text{h}$.
  • Distance traveled by truck: $2.5x\,\text{km}$.
  • Time traveled by passenger car: $2\,\text{h}$.
  • Distance traveled by passenger car: $2(x + 20)\,\text{km}$.
  • Combined distance traveled: $310\,\text{km}$.
  • Formulation & Solution:
  • Formulate equation:       $2.5x + 2(x + 20) = 310$
  • Remove parentheses:       $2.5x + 2x + 40 = 310$
  • Combine like terms:       $4.5x = 270$
  • Solve for $x$:       $x = 60$
  • Truck's speed: $60\,\text{km/h}$.
  • Passenger car's speed: $60 + 20 = 80\,\text{km/h}$.
  • Answer: $60\,\text{km/h}$ for the truck; $80\,\text{km/h}$ for the passenger car.

Exercises and Thought Experiments

• Exercise 156

  • Statement: One number is 4 times greater than a second number. The smaller number is designated as $x$. Express the second number in terms of x$.\n * **Mathematical Expression**: 4x\n\n* **Exercise 157**\n * **Statement**: A total of 62 tulips grow across two flower beds. One flower bed has 6 fewer tulips than the other. Find the number of tulips on each flower bed.\n * **Solution**:\n * Let x be the number of tulips on the smaller bed.\n * The larger bed contains (x + 6) tulips.\n * Set up the total equation:\n      x + (x + 6) = 62\n      2x + 6 = 62\n      2x = 56\n      x = 28\n * Number of tulips on the first bed: 28 tulips.\n * Number of tulips on the second bed: 28 + 6 = 34 tulips.\n * **Answer**: 28 tulips and 34 tulips.\n\n* **Exercise 158 ("Interesting Tasks - Think About It Anyway")**\n * **Statement**: Given that x + y = 13$, where$x$and$y$are natural numbers ($x, y \in \mathbb{N}$), find the values of$x$and$y$for which the expression$x \times y attains its maximum possible value.\n * **Detailed Analysis**:\n * Enumerate all unordered pairs of natural numbers whose sum is 13 and evaluate their product:\n * 1 + 12 = 13 \implies 1 \times 12 = 12\n * 2 + 11 = 13 \implies 2 \times 11 = 22\n * 3 + 10 = 13 \implies 3 \times 10 = 30\n * 4 + 9 = 13 \implies 4 \times 9 = 36\n * 5 + 8 = 13 \implies 5 \times 8 = 40\n * 6 + 7 = 13 \implies 6 \times 7 = 42\n * Observation: The product increases as the difference between the two terms decreases. The maximum product occurs when the integers are closest in value.\n * **Answer**: The expression x \times y$reaches its largest value of 42 when$x = 6, y = 7$or$x = 7, y = 6$.

• Exercise 159 (Oral)

  • Statement: One number is 30 greater than another. The smaller number is denoted as $x$. Express the larger number in terms of x$.\n * **Mathematical Expression**: x + 30\n\n* **Exercise 160 (Oral)**\n * **Statement**: One positive number is 4 times greater than another. The smaller number is denoted as x$. Express the larger number in terms of$x$.
  • Mathematical Expression: $4x$

• Exercise 161

  • Statement: There are $x$ rose bushes growing on one flower bed, and three times as many on a second flower bed. Express the number of rose bushes on the second flower bed in terms of x$.\n * **Mathematical Expression**: 3x$$

• Theoretical Review Question

  • Question: What sequential steps should be followed when solving a problem using an equation?
  • Summary Answer: Identify the unknown with a variable, express other unknowns through that variable, set up an equation using given conditions, solve the linear equation, analyze the resulting value, and write down the final complete answer.