System of equations in Real World Problems 

Systems of equations are powerful tools for solving real-world problems where multiple relationships between variables need to be considered simultaneously. These problems can be found in various fields such as economics, engineering, physics, and even daily life situations.

### Example 1: Business and Economics
**Problem:** A company produces two products, A and B. The company has a limited number of hours available in two departments: manufacturing and packaging. Product A requires 2 hours of manufacturing and 1 hour of packaging. Product B requires 1 hour of manufacturing and 2 hours of packaging. If the company has 100 hours available for manufacturing and 80 hours for packaging, how many units of each product should be produced to maximize the use of available hours?

**System of Equations:**
- \( 2x + y = 100 \)  (Manufacturing hours)
- \( x + 2y = 80 \)   (Packaging hours)

Where:
- \( x \) is the number of units of Product A.
- \( y \) is the number of units of Product B.

**Solution:** Solving this system will tell you how many units of each product to produce.

### Example 2: Mixing Solutions
**Problem:** You have two solutions of saltwater. The first solution is 10% salt, and the second is 25% salt. How many liters of each solution should be mixed to obtain 20 liters of a 15% salt solution?

**System of Equations:**
- \( x + y = 20 \)  (Total volume)
- \( 0.1x + 0.25y = 0.15 \times 20 \)  (Salt concentration)

Where:
- \( x \) is the volume of the 10% solution.
- \( y \) is the volume of the 25% solution.

**Solution:** Solving this system will give the amounts of each solution needed.

### Example 3: Traffic Flow
**Problem:** In a city, two main roads intersect, and the traffic flow needs to be managed. The number of cars entering the intersection from the north (\(x\)) and from the east (\(y\)) must equal the number of cars leaving to the south and west. If 400 cars enter from the north and 300 from the east, how many cars are leaving to the south and west if the intersection can handle only 700 cars at a time?

**System of Equations:**
- \( x + y = 700 \)  (Total cars entering)
- \( x = 400 \)  (Cars entering from the north)
- \( y = 300 \)  (Cars entering from the east)

**Solution:** This system helps in determining how to manage the traffic flow efficiently.

### Steps to Solve Systems of Equations
1. **Define the variables:** Identify what each variable represents in the context of the problem.
2. **Set up the equations:** Translate the problem's conditions into mathematical equations.
3. **Solve the system:** Use substitution, elimination, or matrix methods to solve the equations.
4. **Interpret the solution:** Relate the mathematical solution back to the real-world context.

### Related Questions
1. How can systems of equations be applied in optimizing production schedules?
2. What are some examples of systems of equations in environmental science?
3. How do we model population growth using systems of equations?
4. What role do systems of equations play in financial planning?
5. How can systems of equations help in logistics and supply chain management?

**Tip:** When solving a system of equations from a word problem, always double-check your interpretation of the variables to ensure your equations accurately represent the problem's constraints and goals.

Explore how systems of equations can solve real-world problems in business, chemistry, and traffic flow. Learn step-by-step methods and examples to understand and apply systems of equations effectively.

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