Problem 69 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system for \(x, y,\) and \(z\). At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised \(55 \%\) of the endangered species. Birds accounted for \(0.7 \%\) more than fish, and fish accounted for \(1.5 \%\) more than mammals. What percent of the endangered species came from mammals, birds, and fish?

Step-by-Step Solution

Answer

Mammals: 17.1%, Birds: 19.3%, Fish: 18.6%.

Step 1: Define Variables

Let's define the variables: Let \( x \) represent the percentage of endangered mammals, \( y \) represent the percentage of endangered birds, and \( z \) represent the percentage of endangered fish.

Step 2: Set Up Equations from Problem Statement

From the problem statement, we have the following relationships:1. The sum of the percentages of mammals, birds, and fish is 55%, so: \[ x + y + z = 55 \]2. Birds account for 0.7% more than fish, so: \[ y = z + 0.7 \]3. Fish account for 1.5% more than mammals, so: \[ z = x + 1.5 \]

Step 3: Substitute the Equations and Solve for x

Substitute the expressions for \( y \) and \( z \) from Equations 2 and 3 into Equation 1: \[ x + (z + 0.7) + z = 55 \] Simplifying, we have: \[ x + 2z + 0.7 = 55 \]Substitute \( z = x + 1.5 \) into the equation: \[ x + 2(x + 1.5) + 0.7 = 55 \] Expand and simplify: \[ x + 2x + 3 + 0.7 = 55 \] \[ 3x + 3.7 = 55 \] Subtract 3.7 from both sides: \[ 3x = 51.3 \] Divide by 3: \[ x = 17.1 \]

Step 4: Solve for z

Use the expression \( z = x + 1.5 \) to find \( z \): \[ z = 17.1 + 1.5 = 18.6 \]

Step 5: Solve for y

Use the expression \( y = z + 0.7 \) to find \( y \): \[ y = 18.6 + 0.7 = 19.3 \]

Step 6: Verify the Solution

Check if the percentages add up to 55%: \[ x + y + z = 17.1 + 19.3 + 18.6 = 55 \]This confirms that the solution is correct.

Key Concepts

Endangered Species

In the context of conservation biology, the term "endangered species" refers to groups of organisms that are at risk of extinction due to human activities, environmental changes, or other factors. In our exercise, we focused on the endangered species in the United States.
The statistics presented involved endangered mammals, birds, and fish. These groups make up a significant portion of the species at risk. Understanding their percentages can help prioritize conservation efforts and allocate resources effectively.
Protecting endangered species is crucial for maintaining biodiversity. A healthy ecosystem relies on the balance of different species and their interactions. When one species becomes endangered, it can have a ripple effect on others.

Percentages

Percentages are a way to express proportions as parts of 100, and are used frequently in this exercise to describe the composition of endangered species. Understanding percentages helps simplify complex quantitative information by offering a more intuitive grasp of proportion sizes.
For instance, the problem states that mammals, birds, and fish constitute 55% of the endangered species list. The calculation involved finding the exact percentage each group comprises of this total. By breaking down the percentages, we can get a clearer picture of the relative impact different species groups have on the endangered list.
Utilizing percentages is not only useful in such statistical analysis but also in reporting findings in a more digestible format for policymaking and public awareness.

Solving Equations

Solving equations is a key mathematical skill essential for deriving unknown values from given information. In our exercise, we are tasked with solving a systems of linear equations. This involves finding values of variables that satisfy all equations in the system.
We start by translating words into mathematical expressions. Then, substitute known relationships into equations to simplify and solve them. For instance, we used the relationships between mammal, bird, and fish percentages to form equations.- Equation: \( x + y + z = 55 \)- Birds are 0.7% more than fish: \( y = z + 0.7 \)- Fish are 1.5% more than mammals: \( z = x + 1.5 \)
These equations allow us to find the values of unknown variables through strategic substitution and solving. This systematic approach is crucial for accuracy and understanding.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is fundamental in solving systems of equations like the one presented in this exercise.
We used algebraic manipulation to solve for unknown percentages. By substituting equations into each other, we reduce them to simpler forms. For example, the substitution process combined different relationships to isolate one variable at a time.
Algebra helps not only in solving equations but also in forming them from real-world situations, like determining how much each animal group contributes to the endangered list. Mastery of algebra provides a deeper understanding of how to handle relationships between quantities in various contexts.