Problem 31 - Algebra 1: Concepts and Skills

Question

Use the following information. You are buying vegetables to make a vegetable tray for a party. You buy 10 dollars worth of cauliflower and broccoli. The cauliflower costs 2 dollars per pound and the broccoli costs 1.25 dollars per pound. Write an equation in standard form that represents the different amounts (in pounds) of cauliflower \(C\) and broccoli \(B\) that you could buy.

Step-by-Step Solution

Answer

The equation that represents the different amounts in pounds of cauliflower \(C\) and broccoli \(B\) that could be bought is \(8C + 5B = 40\).

Step 1: Determine the unit price of each vegetable

Given that the cauliflower costs 2 dollars per pound and the broccoli costs 1.25 dollars per pound.

Step 2: Setup the equation

Since the total cost is 10 dollars, and the cost for each vegetable is its respective price per pound times the number of pounds bought, the equation becomes: \(2C + 1.25B = 10\). 'C' represents the weight in pounds of cauliflower bought, and 'B' is the weight in pounds of broccoli bought.

Step 3: Convert the equation into standard form

To convert this equation into standard form, where 'A', 'B', and 'C' are integers and the coefficients 'A' and 'B' are not both zero, we can multiply all terms by 4 to get rid of the fractional coefficient, yielding: \(8C + 5B = 40\) Therefore, the equation in standard form is \(8C + 5B = 40\).

Key Concepts

Standard Form Equation

Linear equations can often be expressed in what is known as the "standard form," which is a format that makes it easy to interpret the relationship between different variables. In our context, the standard form equation looks like this: \[ Ax + By = C \]where:

• '\(A\)' and '\(B\)' are coefficients representing the multiplier effect of their respective variables.
• '\(C\)' is the constant term, representing the total sum or outcome.

It's important that 'A', 'B', and 'C' are integers in the standard form, and more so, 'A' and 'B' should never both be zero because that would nullify the equation. In this exercise, we transform the initial equation \(2C + 1.25B = 10\) into a neat standard form by eliminating the fraction. By multiplying through the equation by 4, we arrive at \(8C + 5B = 40\), ensuring all coefficients remain integers. This final standard form equation lets us easily see what combination of pounds of cauliflower and broccoli meets the $10 constraint.

Unit Price Calculation

Understanding unit price is essential when you are buying multiple items to keep track of costs. Unit price refers to the cost per single unit of measurement. In the exercise, we're given that cauliflower costs 2 dollars per pound, and broccoli costs 1.25 dollars per pound. This information allows us to set up expressions based on weight and costs easily.If we buy \(C\) pounds of cauliflower, the cost is \(2C\); for \(B\) pounds of broccoli, the cost is \(1.25B\). Adding these, our equation \(2C + 1.25B\) helps express the total money spent. This breakdown of unit pricing helps us understand how different amounts of each vegetable combine to meet a specific budget. It provides a clear, concise way to sum up costs in mathematical terms, which is very useful in budgeting and purchasing scenarios.

Mathematical Modeling

Mathematical modeling is all about using equations to represent real-world situations. In this exercise, the goal was to use a linear equation to model the scenario of buying vegetables with a fixed budget. This involves several steps, starting with understanding the "what" – here, it's different quantities of cauliflower and broccoli that add up to $10.The equation \(2C + 1.25B = 10\) models this transaction. Each term represents a cost that contributes to our total spending. To make this model more standard, we adjust it to \(8C + 5B = 40\), helping to clearly delineate each unit’s contribution to overall expenditure. Through mathematical modeling, we bridge real-life purchasing scenarios with understandable mathematical expressions, making it easier to make financial decisions or forecasts by simply plugging in values.