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.
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.
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
To graph the equation 2x + 3y = 12, we can follow these steps:
The equation 2x + 3y = 12 has practical applications in various scenarios:
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:
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.
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.
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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.
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