Problem 64 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. Men aged \(20-29,30-39,\) and 40-49 made up \(78 \%\) of the population at a prison last year. This year, the same age groups made up \(82.08 \%\) of the population. The \(20-29\) age group increased by \(20 \%,\) the 30-39 age group increased by \(2 \%,\) and the \(40-49\) age group decreased to \(\frac{3}{4}\) of their previous population. Originally, the \(30-39\) age group had \(2 \%\) more prisoners than the \(20-29\) age group. Determine the prison population percentage for each age group last year.

Step-by-Step Solution

Answer

The percentages were 32% for 20-29, 34% for 30-39, and 12% for 40-49.

Step 1: Define Variables

Let \( x \), \( y \), and \( z \) represent the percentage of prison population last year for age groups 20-29, 30-39, and 40-49, respectively. Our goal is to find these values.

Step 2: Set Up Equation 1

The total percentage of these three groups last year is given as \(78\%\). Thus, the first equation is \( x + y + z = 78\).

Step 3: Set Up Equation 2

Let's calculate the percentage increase and create the equation for this year. The equation for this year is: \((1.20)x + (1.02)y + (0.75)z = 82.08\).

Step 4: Set Up Equation 3

The problem states that last year, the 30-39 age group had 2% more prisoners than the 20-29 age group, giving us \(y = x + 2\).

Step 5: Substitute to Reduce to Two Equations

Substitute \( y = x + 2 \) from Step 4 into the other two equations. For \( x + y + z = 78\), this becomes \( x + (x + 2) + z = 78 \) or \( 2x + z = 76 \). For \( 1.20x + 1.02y + 0.75z = 82.08 \), substitute \( y \) to get \( 1.20x + 1.02(x + 2) + 0.75z = 82.08 \), simplifying this gives \( 2.22x + 0.75z = 80.04 \). We now have two equations: \( 2x + z = 76 \) and \( 2.22x + 0.75z = 80.04 \).

Step 6: Solve Using Cramer's Rule

Use Cramer's Rule to solve the system:For two equations:1. \( 2x + z = 76 \) 2. \( 2.22x + 0.75z = 80.04 \)The determinant of the coefficient matrix is \(D = \begin{vmatrix} 2 & 1 \ 2.22 & 0.75 \end{vmatrix} = 2 \times 0.75 - 1 \times 2.22 = 1.5 - 2.22 = -0.72\).To solve for \( x \):Replace the first column with the constants:\(D_x = \begin{vmatrix} 76 & 1 \ 80.04 & 0.75 \end{vmatrix} = 76 \times 0.75 - 1 \times 80.04 = 57 - 80.04 = -23.04\).Thus, \( x = \frac{-23.04}{-0.72} = 32\).To solve for \( z \):Replace the second column with the constants:\( D_z = \begin{vmatrix} 2 & 76 \ 2.22 & 80.04 \end{vmatrix} = 2 \cdot 80.04 - 76 \cdot 2.22 = 160.08 - 168.72 = -8.64\).Thus, \( z = \frac{-8.64}{-0.72} = 12\).

Step 7: Calculate y Using Equation from Step 4

Substitute \(x = 32\) into \(y = x + 2\) from Step 4 to find \(y\).So, \(y = 32 + 2 = 34\).

Step 8: Verify the Solution

Verify the solution fits all equations:- Check \( x + y + z = 78 \):\( 32 + 34 + 12 = 78 \) (Valid)- Check \( 1.20x + 1.02y + 0.75z = 82.08 \):\( 1.20 \times 32 + 1.02 \times 34 + 0.75 \times 12 = 38.4 + 34.68 + 9 = 82.08 \) (Valid).

Step 9: State the Solution

The percentages of the prison population for each age group last year were \( 32\% \) for 20-29, \( 34\% \) for 30-39, and \( 12\% \) for 40-49.

Key Concepts

System of Linear Equations

When we talk about a system of linear equations, we're referring to a set of equations that involve two or more variables. In these problems, you're usually looking for solutions that satisfy all the equations at the same time. Imagine you have puzzle pieces that need to perfectly fit together.

In our exercise, the puzzle pieces are the percentages of different age groups in a prison population. We have three different ages—20-29, 30-39, and 40-49—and each age has a percentage that we want to find. We represent these unknown percentages as variables, namely \(x\), \(y\), and \(z\). Each equation in our problem provides a specific relationship between these variables.

For instance:

• The sum of percentages for all three age groups was given as 78% last year.
• There was a change in percentages this year that affects them, expressed in another equation.
• The percentage of one age group was connected to another by a specific difference.

Using Cramer's Rule, these equations are the essential tools to solve for the unknowns and understand how different age groups contribute to the total population percentage.

Determinant

Understanding the determinant is crucial when using methods like Cramer's Rule. A determinant is a special number that can be calculated from a square matrix—a grid of numbers lined up in rows and columns.

In the context of solving linear equations, the determinant helps you find out if the system has a unique solution. If the determinant is not zero, this means you can find a unique solution for the variables.

In our exercise, the coefficient matrix of the system of equations is \[\begin{vmatrix} 2 & 1 \ 2.22 & 0.75 \end{vmatrix}\] This means the first row is made of the coefficients 2 and 1, and the second row is composed of 2.22 and 0.75.

You calculate the determinant by multiplying diagonals and subtracting the products:\[\text{Determinant} = (2 \times 0.75) - (1 \times 2.22) = 1.5 - 2.22 = -0.72\]
Since the determinant is not zero, our system has a unique solution, making it possible to find the exact values for \(x\), \(y\), and \(z\).

Prison Population Percentages

Analyzing prison population percentages involves understanding how the proportions of different groups within a population change over time. In our problem, we look at specific age groups and how their respective percentages have adjusted from one year to the next.

This context involves setting up equations that express these relationships clearly. We have three age groups with given percentage values that changed due to:

• An increase in the 20-29 age group's population by 20%
  
• An increase in 30-39 by 2%
  
• A decrease in 40-49 down to three-quarters of their original proportions

Our task is to find the starting percentages from last year, solving a kind of puzzle to end up with the numbers that add up to the current year's known situation—82.08%.
Using all the given relationships, a system is established to calculate each group's past contributions to the prison population correctly.

Age Group Analysis

Age group analysis in this context is evaluating and interpreting data about different age categories within a population. It's significant for understanding trends and changes over time, especially when there's a question about how different factors might be influencing the proportions of these groups.

When you think about these age groups in a prison setting, you're essentially analyzing the age distribution and changes. The exercise specifically points out how each group's percentage changed:

• The youngest group's percentage increased significantly.
• The middle group's percentage had a slight growth.
• The oldest group's percentage decreased notably by being reduced to three-quarters of its numbers.

This analysis helps crack the problem using the information from the equations and determining past percentages.
Age group analysis enables better understanding of population dynamics, leading to informed decisions and strategies in managing these populations.