Problem 30 - Algebra 2

Question

Use the table showing state populations. \(\begin{array}{|c|c|c|c|c|}\hline 1 & {\text { California }} & {25,484,000} & {567,000} \\ \hline 2 & {\text { Texas }} & {22,118,000} & {447,000} \\\ \hline 3 & {\text { New York }} & {19,190,000} & {70,000} \\ \hline 4 & {\text { Florida }} & {17,019,000} & {304,000} \\ \hline 5 & {\text { lllinois }} & {12,653,000} & {80,000} \\ \hline\end{array}\) Write equations that represent the populations of Florida and New York \(x\) years after 2003. Assume that both states continue to gain the same number of residents every year. Let \(y\) equal the population.

Step-by-Step Solution

Answer

Florida: \(y = 17,019,000 + 304,000x\); New York: \(y = 19,190,000 + 70,000x\)

Step 1: Identify Initial Population and Annual Increase

Start by identifying the initial population and the annual increase for each state. For Florida, the initial population in 2003 is 17,019,000 and it increases by 304,000 residents each year. For New York, the initial population in 2003 is 19,190,000 and it increases by 70,000 residents each year.

Step 2: Define Variables

Introduce variables for the equations. Let \(y\) represent the population and \(x\) represent the number of years after 2003. The equations for both states will use these variables.

Step 3: Construct the Equation for Florida

Using the information from Step 1, the equation for Florida's population can be set up as \(y = 17,019,000 + 304,000x\). This equation represents the population \(y\) as a function of years \(x\) after 2003.

Step 4: Construct the Equation for New York

Similarly, based on the provided data, New York's population equation is \(y = 19,190,000 + 70,000x\). This formula calculates the New York population \(y\) based on the number of years \(x\) after 2003.

Step 5: Review and Confirm Equations

Ensure both equations accurately reflect the population data from the table: Florida's equation is \(y = 17,019,000 + 304,000x\) and New York's equation is \(y = 19,190,000 + 70,000x\). Double-check calculations and variable placements to confirm correctness.

Key Concepts

Population Growth

Population growth is a concept that helps us understand how populations increase over time. It can result from various factors, but in this exercise, we focus on linear growth, where each year sees a fixed number of new residents added to the population.
Florida and New York provide perfect examples. In 2003, Florida had a population of 17,019,000, and it gains 304,000 new residents each year. On the other side, New York had a population of 19,190,000 and an annual increase of 70,000 residents.
These steady increases result in predictable, linear population growth. It's helpful in planning and resource allocation, knowing how many people will reside in a state in the future.

Algebraic Expressions

Algebraic expressions are mathematical phrases that link numbers and variables with operational symbols. In our exercise, they connect population and time through changes in state growth.
To represent Florida's population, we use the equation: \( y = 17,019,000 + 304,000x \). Here, \( y \) is the state's population, while \( x \) is the number of years since 2003. The number 17,019,000 matches the base population, and 304,000 is the yearly population increment.
For New York, the expression is \( y = 19,190,000 + 70,000x \). Each term reflects the population growth at constant intervals of years. Algebra simplifies these relationships, providing a clear way to model changes over time.

Word Problems

Word problems are scenarios where mathematical operations are applied to real-world situations. They help you to translate a story or context into numbers and equations.
In the case of our exercise, we've integrated the concept of population growth into a mathematical framework. Initially, you gather the critical figures: initial population and yearly growth. Then, you craft an algebraic equation representing Florida and New York's population growth over time.
By solving or analyzing these equations, you gain insights into how the population will evolve. This fits broader applications, such as predicting resource needs or urban planning, making sure every equation or formula ties directly with the word scenario presented.