Understanding the Equation 2x + 3y = 12: A Comprehensive Guide

The equation 2x + 3y = 12 is a linear equation in two variables, x and y. It represents a straight line on a coordinate plane. This article aims to provide a comprehensive understanding of this equation, exploring its properties, applications, and significance.

Breaking Down the Equation

2x: This term represents twice the value of x. The coefficient 2 indicates that the value of x is multiplied by 2.

3y: Similarly, this term represents three times the value of y. The coefficient 3 indicates that the value of y is multiplied by 3.

12: This is the constant term, which represents a fixed value. It does not vary with the values of x and y.

Solving for x and y

To solve the equation for x or y, we can isolate the variable on one side of the equation.

Solving for x:

2x + 3y = 12
2x = 12 - 3y
x = (12 - 3y) / 2

Solving for y:

2x + 3y = 12
3y = 12 - 2x
y = (12 - 2x) / 3

Graphing the Equation

To graph the equation 2x + 3y = 12, we can follow these steps:

1. Find the intercepts:
   • Set x to 0: 2(0) + 3y = 12 -> y = 4 (y-intercept)
   • Set y to 0: 2x + 3(0) = 12 -> x = 6 (x-intercept)
2. Plot the intercepts: Mark the points (0, 4) and (6, 0) on the coordinate plane.
3. Draw the line: Connect the two intercepts with a straight line.

Applications in Everyday Life

The equation 2x + 3y = 12 has practical applications in various scenarios:

• Mixing paint: If you have two types of paint, one with a volume of x gallons and the other with a volume of y gallons, and you need a total of 12 gallons of a mixture, then the equation would be: 2x + 3y = 12.
• Budgeting: If you earn $2x per hour and work y hours per week, and your weekly expenses are $12, then the equation would be: 2x + 3y = 12.
• Recipe scaling: If a recipe calls for 2 cups of flour and 3 cups of milk, and you want to double the recipe, then the equation would be: 2(2x) + 3(2y) = 12.

Significance in Mathematics

The equation 2x + 3y = 12 is a special case of the general linear equation Ax + By = C, where A, B, and C are constants. Linear equations are fundamental in mathematics and have applications in various fields, including:

• Physics: Modeling motion, forces, and other physical phenomena.
• Economics: Analyzing supply and demand, market equilibrium, and other economic models.
• Computer science: Solving algorithms, designing circuits, and creating computer graphics.

Real-World Stories and Lessons

Story 1: A farmer has 20 acres of land that he wants to plant with wheat and corn. He knows that each acre of wheat yields 2 tons, while each acre of corn yields 3 tons. He wants to harvest a total of 12 tons. How many acres of each should he plant?

Solution: The equation would be: 2x + 3y = 12, where x is the number of acres planted with wheat and y is the number of acres planted with corn. Solving for x and y, we get x = 6 (acres of wheat) and y = 2 (acres of corn).

Lesson: Breaking down a problem into a mathematical equation can help us find practical solutions.

Story 2: A clothing store is having a sale on shirts and pants. Shirts cost $2 each, while pants cost $3 each. If a customer spends $12, how many shirts and pants did they buy if they bought an equal number of each?

Solution: Let x be the number of shirts and y be the number of pants. The equation would be: 2x + 3y = 12. Since they bought an equal number of each, x = y. Substituting this into the equation, we get x = 2 (shirts) and y = 2 (pants).

Lesson: Setting up an equation with equal variables can help us solve problems involving ratios.

Story 3: A company's revenue in millions of dollars is given by the equation R = 2x + 3y, where x is the number of units sold and y is the price per unit. In January, they sold 4 units at a price of $3 per unit. In February, they sold 5 units at a price of $2 per unit. What was their total revenue for January and February?

Solution: For January, the revenue is R = 2(4) + 3(3) = $18 million. For February, the revenue is R = 2(5) + 3(2) = $14 million. Their total revenue is $32 million.

Lesson: Linear equations can be used to model real-world data and calculate outcomes in business or finance.

Tips and Tricks

• Simplify the equation: If possible, combine like terms or factor out common factors to make the equation easier to solve.
• Use graphing software: Graphing calculators or online tools can help you visualize the equation and find its intercepts and slope.
• Substitute known values: If you know the value of one variable, you can substitute it into the equation to solve for the other variable.
• Check your solution: Once you have found a solution, substitute it back into the original equation to make sure it satisfies the equation.

Why It Matters

The equation 2x + 3y = 12 is a fundamental concept in mathematics that has important applications in everyday life and various academic disciplines. It provides a framework for understanding linear relationships and solving a wide range of problems. By understanding this equation and its properties, we can gain valuable analytical and problem-solving skills.

Benefits of Understanding the Equation

• Facilitates problem-solving in real-world scenarios involving proportions and ratios.
• Enhances critical thinking and analytical reasoning skills.
• Provides a foundation for understanding more complex mathematical equations and models.
• Supports applications in fields such as economics, physics, engineering, and computer science.
• Develops a foundation for further mathematical studies and research.

Comparison of Pros and Cons

Pros:

• Simple and easy to understand.
• Versatile and applicable in various contexts.
• Provides a structured approach to problem-solving.
• Encourages logical thinking and reasoning abilities.

Cons:

• May be limited in solving more complex equations or systems.
• Requires careful attention to variable manipulation and calculation.
• May require additional mathematical techniques to extend its applicability.
• Can be challenging to visualize for some individuals who do not have a strong spatial understanding.

Conclusion

The equation 2x + 3y = 12 is a foundational building block in mathematics with far-reaching applications. By embracing its concepts and principles, we can enhance our problem-solving capabilities, develop our understanding of the world around us, and unlock new possibilities in various fields.