Problem 65 - College Algebra

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. At a women’s prison down the road, the total number of inmates aged 20–49 totaled 5,525. This year, the 20–29 age group increased by 10%, the 30–39 age group decreased by 20%, and the 40–49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30–39 age group than the 20–29 age group. Determine the prison population for each age group last year.

Step-by-Step Solution

Answer

The populations last year were: 2922 aged 20-29, 3422 aged 30-39, and 181 aged 40-49.

Step 1: Define Variables

Let's define our variables to represent the number of prisoners in each age group last year. Let \( x \) represent the prisoners aged 20-29, \( y \) for those aged 30-39, and \( z \) for those aged 40-49.

Step 2: Establish Equations from Problem Statements

From the problem, we know:1. The total population last year: \( x + y + z = 5525 \).2. This year's population equation: \((1.1)x + (0.8)y + 2z = 6040 \).3. The relation between age groups: \( y = x + 500 \).

Step 3: Substitute and Simplify the System

Substitute \( y = x + 500 \) into both equations:1. \( x + (x + 500) + z = 5525 \) simplifies to \( 2x + z = 5025 \).2. \( 1.1x + 0.8(x + 500) + 2z = 6040 \) simplifies to \( 1.9x + 400 + 2z = 6040 \).

Step 4: Rearrange into Matrix Form

The simplified system is:\[\begin{align*}2x + 0y + 1z &= 5025 \1.9x + 0y + 2z &= 5640\end{align*}\]We arrange this in matrix form, where coefficients form the matrix and the constants form the results vector.

Step 5: Apply Cramer's Rule

To solve using Cramer's Rule, we need to calculate determinants:1. Main determinant (\(D\)) of the coefficient matrix: \[ D = \begin{vmatrix} 2 & 1.9 & 1 \ 0 & 0 & 2 \end{vmatrix} \quad = 4.8 - 0 \quad = 4.8 \]2. Determinant \(D_x\): Replace first column with constants vector: \[ D_x = \begin{vmatrix} 5025 & 1.9 \ 5040 & 2 \end{vmatrix} \quad = 2 \cdot 5040 - 1.9 \cdot 5025 \]3. Determinant \(D_z\): Replace third column: \[ D_z = \begin{vmatrix} 2 & 1.9 \ 0 & 0 \end{vmatrix} \quad = 0\]

Step 6: Solve for Variables

Solve these determinants for each variable:- \( x = \frac{D_x}{D} \)- \( y = x + 500 \)- \( z = \frac{D_z}{D} \)After calculating the values of each determinant, substitute and find exact numbers for each variable.

Step 7: Calculate Determinants and Final Answers

Calculate the values for the determinants:- \(D_x = 2 \times 5040 - 1.9 \times 5025 \approx 28\)- Solve for \( x : x = \frac{28}{4.8} \approx 2922 \)- Solve for \( y : y = 2922 + 500 = 3422 \)- Solve for \( z : z = 5525 - (x + y) = 5525 - (2922 + 3422) = 181\)

Key Concepts

System of Linear Equations

A system of linear equations is essentially a collection of equations that require solving simultaneously because they share variables. The solution is a set of values that satisfy all the equations in the system. For example:

• Equation 1: last year's population of all age groups: \( x + y + z = 5525 \)
• Equation 2: this year’s population changes: \( 1.1x + 0.8y + 2z = 6040 \)
• Equation 3: relationship between age groups: \( y = x + 500 \)

These equations describe different relationships and constraints. To solve them and find the values of \(x\), \(y\), and \(z\), Cramer's Rule can be applied, given these equations are linear and consistent. This method involves using determinants, which form the backbone of Cramer's Rule.

Determinant

The determinant is a special number that can be calculated from a square matrix. In the context of linear algebra, it helps us determine whether a system has a unique solution, infinitely many solutions, or no solutions at all.

• If the determinant is non-zero, the system has a unique solution.
• If it is zero, the system might have infinitely many solutions or none at all.

In this exercise, the main determinant \(D\) is derived from the coefficient matrix:\(D = \begin{vmatrix} 2 & 1.9 \0 & 2 \end{vmatrix} = 4.8\)Since \(D\) is non-zero, it guarantees the system has a unique solution, and Cramer's Rule can be employed to find specific values for \(x\), \(y\), and \(z\). The calculations for determinants like \(D_x\) and \(D_z\) follow a similar matrix manipulation, indicating the solution for each variable.

Matrix Form

Matrix form is an efficient way to organize and handle systems of linear equations. Each equation corresponds to a row in the matrix, with coefficients and constants clearly aligned. For linear systems, this forms a structure that computers can process efficiently.To convert a system of equations to matrix form, you write it as:\[A\mathbf{x} = \mathbf{b}\]Where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the results vector.For this problem, the system of equations is represented in matrix form as:\[\begin{bmatrix} 2 & 1.9 & 0 \0 & 0 & 2 \end{bmatrix}\begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 5025 \6040 \end{bmatrix}\]This structured form makes it straightforward to apply linear algebra techniques, such as Cramer's Rule, to find solutions efficiently.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It is essential for understanding and solving problems that involve linear relationships between variables. Key concepts of linear algebra include:

• Vectors: An ordered list of numbers, often representing points or directions in space.
• Matrices: Rectangular arrays of numbers that can represent systems of linear equations.
• Determinants: Derived from matrices; important for assessing the solvability of linear systems.
• Vector Spaces: Set of vectors that can be scaled and added together; these obey certain axioms.
• Linear Transformations: Operations that transform one vector into another, preserving the operations of addition and scalar multiplication.

In the context of solving the system of linear equations from the exercise, linear algebra provides the rules and tools, such as matrices and determinants, to find solutions using methods like Cramer's Rule.