Question

Use the following information. You are buying 20 dollars worth of bird seed that consists of two types of seed. Thistle seed attracts finches and costs 2 dollars per pound. Dark oil sunflower seed attracts many kinds of song birds and costs 1.50 dollars per pound. Write an equation that represents the different amounts of 2 dollars thistle seed \(x\) and 1.50 dollars dark oil sunflower seed \(y\) that you could buy.

Short answer

The equation representing the different amounts of thistle seed \(x\) and dark oil sunflower seed \(y\) that you could buy is \(2x + 1.5y = 20\).

Step-by-step solution

Define the Variables

Define variables to represent the amounts of the two types of bird seed. Let \(x\) represent the amount of thistle seed in pounds and \(y\) represent the amount of sunflower seed in pounds.

Identify the Cost of Each Type of Seed

It's given in the problem that the cost of thistle seed is 2 dollars per pound and the cost of sunflower seed is 1.50 dollars per pound. So, the total cost for \(x\) pounds of thistle seed is \(2x\) dollars and the total cost for \(y\) pounds of sunflower seed is \(1.5y\) dollars.

Formulate the Equation

The total cost for \(x\) pounds of thistle seed and \(y\) pounds of sunflower seed is 20 dollars. So, we can create an equation representing this situation. It would be \(2x + 1.5y = 20\). This equation represents amounts of thistle seed \(x\) and sunflower seed \(y\) that could be bought with 20 dollars.

Key concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and are used to represent real-world situations mathematically. They consist of variables, numbers, and operations (such as addition and multiplication) combined in a way that allows for computation and manipulation. For example, in the context of purchasing bird seed, where thistle seed costs 2 dollars per pound and dark oil sunflower seed costs 1.50 dollars per pound, the expression to calculate the total cost would involve the quantities of each type of seed bought multiplied by their respective costs. To easily understand, think of algebraic expressions as a recipe that tells you how to mix different ingredients—in this case, the types and amounts of bird seed—to achieve a desired result, which is staying within budget. So an algebraic expression is like a mathematical recipe for real-life scenarios.

When creating algebraic expressions, it's crucial to maintain proper mathematical grammar. Just as sentences have a structure, so do algebraic expressions. For instance, the expression for the total cost of the bird seed is written as a sum of the individual costs: the cost for thistle seed (\(2x\) dollars) plus the cost for dark oil sunflower seed (\(1.5y\) dollars), reflecting the combined expense for both types of seed.

Variable Representation

Variables in algebra are symbols (usually letters) that stand in for unknown or changeable values. They are the placeholders in the mathematical world and are incredibly powerful in representing the quantities in a problem. In our bird seed example, we're using two variables, \(x\) and \(y\), to represent the unknown amounts of thistle seed and sunflower seed, respectively.

Proper variable representation allows us to create equations and solutions that can adapt to different values. This means that once we have our equation, we can change the values of \(x\) and \(y\) to see how different quantities of seed affect the total cost. It's similar to adjusting the quantities in a recipe—if you know you want more cookies, you adjust the recipe accordingly. In math, variables let you 'adjust the recipe' of your problem to explore various scenarios. When choosing variables, it's helpful to select symbols that are meaningful and related to the problem at hand, making it easier to remember and understand their roles.

Equation Formulation

Equation formulation is the process of creating a mathematical statement that equates two expressions, typically representing a balance between two conditions. For our bird seed scenario, equation formulation involves setting the total cost of seed equal to the amount of money you have to spend. The key to forming correct equations is to carefully translate the conditions of your real-world problem into mathematical language.

To craft an equation, start by clearly defining what each term represents, as was done with \(x\) and \(y\) for the pounds of bird seed. Grasp the relationship between these terms—in this case, the costs associated with each type of seed. Then, consider the overall condition that must be met, such as not exceeding the 20-dollar budget. Finally, combine these components into an equation that captures the essence of the problem: \(2x + 1.5y = 20\). This step is much like solving a puzzle; each piece must fit precisely to complete the picture. By following the rules of algebra and understanding the relationships in your problem, you can create an equation that serves as a model to find solutions.