Solve System of Equations - Word Problem - Algebra-1

1. Fundamental Concepts

System of Equations: A set of two or more equations containing the same unknowns, used to describe the relationships between different quantities in the same problem.
Word Problem: A problem that describes a real-world situation through text, requiring the conversion of textual information into mathematical equations to solve for unknowns.
Core Goal: To find the values of the unknowns that satisfy all equations (i.e., the solution to the system), which must conform to the real-world context (e.g., the number of people or length must be non-negative).
Variables: Letters (such as x, y) used to represent unknown quantities in the problem, with each variable's actual meaning clearly defined (e.g., "Let x be the unit price of apples").

2. Key Concepts

Equal Relationships: Implied "equality" relationships in word problems, which are the basis for formulating equations. Common types include:
- Total quantity: e.g., "Total number of people = number of boys + number of girls";
- Price calculation: e.g., "Total price = unit price × quantity";
- Proportion relationship: e.g., "Concentration = mass of solute ÷ mass of solution";
- Travel problems: e.g., "Distance = speed × time".
Types of Systems of Equations:
- System of linear equations in two variables: Contains 2 unknowns, with each unknown raised to the first power (e.g., \(\begin{cases}x + y = 5 \\ 2x - y = 1\end{cases}\));
- Systems with multiple variables: Contains 3 or more unknowns (e.g., system of linear equations in three variables).
Reasonableness of Solutions: After solving, verify whether the solution conforms to practical significance (e.g., "The number of people cannot be negative" or "Time cannot be negative").

3. Examples

Easy Difficulty

Problem: A fruit basket contains a total of 10 apples and bananas. There are 2 more apples than bananas. How many apples and bananas are there?

Solution:

Let x be the number of apples and y be the number of bananas.
Formulate the system of equations: \(\begin{cases}x + y = 10 \\ x - y = 2\end{cases}\)
Solve: Add the two equations to get \(2x = 12\), so \(x = 6\). Substitute \(x = 6\) into the first equation to get \(y = 4\).
Conclusion: There are 6 apples and 4 bananas.

Medium Difficulty

Problem: A store sells 3 pens and 2 notebooks for a total of 34 yuan, and 2 pens and 3 notebooks for a total of 31 yuan. Find the unit price of 1 pen and 1 notebook.

Solution:

Let x be the unit price of a pen (in yuan) and y be the unit price of a notebook (in yuan).
Formulate the system of equations: \(\begin{cases}3x + 2y = 34 \\ 2x + 3y = 31\end{cases}\)
Solve: Multiply the first equation by 2: \(6x + 4y = 68\); multiply the second equation by 3: \(6x + 9y = 93\). Subtract the two equations to get \(5y = 25\), so \(y = 5\). Substitute \(y = 5\) into the first equation to get \(x = 8\).
Conclusion: The unit price of a pen is 8 yuan, and the unit price of a notebook is 5 yuan.

Hard Difficulty

Problem: The distance between places A and B is 300 km. Car A starts from place A, and Car B starts from place B, moving towards each other. The speed of Car A is 10 km/h faster than that of Car B. The two cars meet after 2 hours of travel. After meeting, Car A continues to travel to place B at its original speed, and Car B continues to travel to place A at its original speed. Find the total time for Car A to travel from place A to place B and the total time for Car B to travel from place B to place A.

Solution:

Let the speed of Car B be x km/h and the speed of Car A be y km/h.
From "The speed of Car A is 10 km/h faster than that of Car B", we get the equation: \(y = x + 10\).
Since the two cars are moving towards each other and meet after 2 hours, with the total distance between A and B being 300 km, according to "Distance = speed × time" and "Total distance = distance traveled by Car A in 2 hours + distance traveled by Car B in 2 hours", we get the equation: \(2x + 2y = 300\).
This gives the system of equations: \(\begin{cases}y = x + 10 \\ 2x + 2y = 300\end{cases}\)
Solve: Substitute \(y = x + 10\) from the first equation into the second equation \(2x + 2y = 300\), we get \(2x + 2(x + 10) = 300\). Expanding the brackets: \(2x + 2x + 20 = 300\). Combining like terms: \(4x + 20 = 300\). Rearranging: \(4x = 280\), so \(x = 70\).
Substitute \(x = 70\) into \(y = x + 10\), we get \(y = 70 + 10 = 80\). That is, the speed of Car B is 70 km/h, and the speed of Car A is 80 km/h.
Calculate the total time for Car A to travel from A to B: Using "Time = distance ÷ speed", with a total distance of 300 km and a speed of 80 km/h, the total time is \(300 \div 80 = 3.75\) hours (i.e., 3 hours and 45 minutes).
Calculate the total time for Car B to travel from B to A: With a speed of 70 km/h and a total distance of 300 km, the total time is \(300 \div 70 = \frac{30}{7} \approx 4.29\) hours.
Conclusion: The total time for Car A to travel from place A to place B is 3.75 hours, and the total time for Car B to travel from place B to place A is approximately 4.29 hours.

4. Problem-Solving Techniques

Understanding the Problem and Setting Variables:
- Read the problem thoroughly and mark key information (such as "total", "more than", "speed", etc.);
- When setting variables, clearly define the actual quantity each letter represents (e.g., "Let x be the number of adult tickets"), with the number of variables matching the number of equal relationships.
Formulating Equations:
- Convert text into mathematical expressions based on "equal relationships". For example:
  - "The quantity of A is twice that of B" → \(A = 2B\);
  - "Buying A and B costs 50 yuan, with A at 3 yuan each and B at 2 yuan each" → \(3A + 2B = 50\).
Solving the System of Equations:
- Substitution method: Express one variable in terms of the other from one equation and substitute it into the other equation;
- Elimination method: Multiply equations by coefficients to make the coefficients of a certain variable opposite or equal, then add or subtract to eliminate that variable.
Verification and Answering:
- Check if the solution satisfies all equations;
- Verify if the solution conforms to reality (e.g., the number of people is an integer, time is positive);
- Answer the problem in words (e.g., "There are 5 apples").
Specialized Breakthroughs for Common Problem Types:
- Travel problems: Distinguish between meeting scenarios (sum of distances = total distance) and chasing scenarios (difference in distances = initial distance);
- Profit problems: Profit = selling price - cost, total profit = profit per unit × quantity;
- Concentration problems: The mass of solute remains unchanged before and after dilution/mixing.