Problem 17 - Calculus for Biology and Medicine

Question

Zach wants to buy fish and plants for his aquarium. Each fish costs \(\$ 2.30 ;\) each plant costs \(\$ 1.70 .\) He buys a total of 11 items and spends a total of \(\$ 21.70 .\) Set up a system of linear equations that will allow you to determine how many fish and how many plants Zach bought, and solve the system.

Step-by-Step Solution

Answer

Zach bought 5 fish and 6 plants.

Step 1: Define Variables

Let's define the variables needed to create the equations. Let \( f \) represent the number of fish Zach buys, and \( p \) represent the number of plants he buys.

Step 2: Create the Equation for Total Items

Since Zach buys a total of 11 items, we can express this relationship with the equation: \[ f + p = 11 \]

Step 3: Create the Equation for Total Cost

Zach spends a total of \\(21.70. Each fish costs \\)2.30 and each plant costs \$1.70. We can write this equation as: \[ 2.30f + 1.70p = 21.70 \]

Step 4: Solve the System of Equations

We have the system of equations: \[ f + p = 11 \] \[ 2.30f + 1.70p = 21.70 \] Solve the first equation for \( p \): \[ p = 11 - f \]Substitute \( p = 11 - f \) into the second equation:\[ 2.30f + 1.70(11 - f) = 21.70 \] Simplify and solve for \( f \):\[ 2.30f + 18.70 - 1.70f = 21.70 \] \[ 0.60f + 18.70 = 21.70 \] \[ 0.60f = 3.00 \] \[ f = \frac{3.00}{0.60} = 5 \] Substitute \( f = 5 \) back into \( p = 11 - f \):\[ p = 11 - 5 = 6 \]

Step 5: Verify the Solution

Verify that the solution satisfies both the total number of items and total cost. \( f + p = 5 + 6 = 11 \) items, which is correct. Check the cost:\[ 2.30 \times 5 + 1.70 \times 6 = 11.50 + 10.20 = 21.70 \] Both equations are satisfied, confirming the solution is correct.

Key Concepts

Understanding Variables in Equations

In a system of linear equations, variables are key components representing unknown values that we need to find. In our example with Zach, we define two essential variables:

• Variable \( f \): Represents the number of fish Zach buys.
• Variable \( p \): Represents the number of plants Zach buys.

By using these variables, we can create equations that symbolically represent real-world situations. Variables give us a flexible framework to work with mathematical expressions and solve unknowns, like how many fish or plants Zach bought. It's important to clearly define these variables before diving into solving the equations as it sets a foundation for understanding the problem. Whenever you encounter a word problem, look for quantities that require disclosing unknowns and assign them variables. This step is crucial as it transforms the narrative into a quantifiable scenario.

Approach to Solving Equations

Solving a system of linear equations involves determining the value of variables that satisfy all provided equations. In Zach's challenge, we set up two equations:

• Total Items Equation: \( f + p = 11 \), indicating the total number of items (fish and plants).
• Total Cost Equation: \( 2.30f + 1.70p = 21.70 \), estimating the total expenditure on these items.

To solve these, we use substitution, a method where one equation is solved for one variable, then substituted into the other equation. For instance, we solve for \( p \) from the first equation and substitute it in the second. This leads us to a single equation with one variable:\[ 2.30f + 1.70(11 - f) = 21.70 \]Simplifying, we find: \[ 0.60f = 3.00 \]Finally, we obtain \( f \), and use it to find \( p \). Solving each equation confirms the values satisfy both conditions provided, verifying the accuracy of our solution. This process shows how intertwining equations reflect dependencies of different aspects of a situation.

Exploring Cost Analysis

Cost analysis in algebra involves comparing values to determine totals and make informed decisions. In Zach's scenario, cost analysis means understanding how each purchase contributes to the overall cost:

• Each fish costs \\(2.30, impacting the total cost with every additional fish purchased.
• Each plant costs \\)1.70, similarly affecting the total expense.

The equation \( 2.30f + 1.70p = 21.70 \) expresses how the cost of fish and plants adds up to the total budget Zach spends. By examining these relationships, we see how changing the quantity of one item adjusts total expenses accordingly. Understanding cost analysis allows us to evaluate spending relative to budget constraints and optimize buying decisions.This equation leads to better insights into how each dollar is spent and provides a fiscal overview of where to allocate resources efficiently to stay within a budget. Thus, knowing how to perform cost analysis not only solves problems but also manages real-world financial situations.