Problem 11 - Precalculus

Question

Use Cramer’s Rule to solve the system of linear equations. $$ \left\\{\begin{aligned} x+y &=8000 \\ 0.03 x+0.05 y &=250 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 7500 \) and \( y = 500 \).

Step 1: Write the System in Matrix Form

Convert the given system of equations into a matrix form. The system is:\[ \begin{aligned} x + y &= 8000 \ 0.03x + 0.05y &= 250 \end{aligned} \] This can be written as \( AX = B \), where \( A = \begin{bmatrix} 1 & 1 \ 0.03 & 0.05 \end{bmatrix} \), \( X = \begin{bmatrix} x \ y \end{bmatrix} \), and \( B = \begin{bmatrix} 8000 \ 250 \end{bmatrix} \).

Step 2: Find Determinant of A

Compute the determinant of matrix \( A \). The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). For matrix \( A \), \( |A| = (1)(0.05) - (0.03)(1) = 0.05 - 0.03 = 0.02 \).

Step 3: Formulate Determinant for x

Create matrix \( A_x \) by replacing the first column of \( A \) with \( B \). Thus \( A_x = \begin{bmatrix} 8000 & 1 \ 250 & 0.05 \end{bmatrix} \). Compute the determinant, \( |A_x| = (8000)(0.05) - (1)(250) = 400 - 250 = 150 \).

Step 4: Formulate Determinant for y

Create matrix \( A_y \) by replacing the second column of \( A \) with \( B \). Thus \( A_y = \begin{bmatrix} 1 & 8000 \ 0.03 & 250 \end{bmatrix} \). Compute the determinant, \( |A_y| = (1)(250) - (0.03)(8000) = 250 - 240 = 10 \).

Step 5: Solve for x and y Using Cramer's Rule

Apply Cramer's Rule, which states: \( x = \frac{|A_x|}{|A|} \) and \( y = \frac{|A_y|}{|A|} \). Therefore, \( x = \frac{150}{0.02} = 7500 \) and \( y = \frac{10}{0.02} = 500 \).

Key Concepts

Linear Equations

Linear Equations are rules that involve constants, variables, and arithmetic operations, like addition and multiplication. These equations often show how different variables are connected.
In the context of systems of linear equations, you have multiple equations that must be solved together to find values of variables.
For example, if we have two linear equations with two variables, such as:

• \(x + y = 8000\)
• \(0.03x + 0.05y = 250\)

Here, a solution means finding specific values of \(x\) and \(y\) that make both equations true simultaneously. These linear equations can have exactly one solution, infinitely many solutions, or no solution at all depending on how they are set up.

Matrix Determinant

A Matrix Determinant is a special value that can be calculated from a square matrix. It provides insightful details about the matrix, like if it's invertible or the orientation of vectors when transformed by the matrix.
The determinant for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as:

• \(ad - bc\)

In our problem, the given matrix \(A\) is:

• \( \begin{bmatrix} 1 & 1 \ 0.03 & 0.05 \end{bmatrix} \)

The determinant is found as \( |A| = (1 \times 0.05) - (0.03 \times 1) = 0.02 \). Having a nonzero determinant (here it's 0.02) means that the matrix is invertible and the system of equations has a unique solution.

System of Equations

A System of Equations is a collection of two or more equations with the same set of unknowns. The goal is to find the values of the variables that satisfy all equations in the system.
In our example, the system consists of the following equations:

• \(x + y = 8000\)
• \(0.03x + 0.05y = 250\)

To find a solution, the equations can be solved using various methods such as substitution, elimination, and matrix operations, with Cramer's Rule being one of the matrix-based methods. Cramer's Rule is particularly effective in cases where the coefficient matrix has a non-zero determinant, ensuring that a unique solution exists.

Matrix Formulation

Matrix Formulation is a method of organizing systems of linear equations into a compact form using matrices. This approach allows the use of various matrix operations, such as finding determinants and inverses, to solve systems of equations efficiently.
In our case, the system of equations involving \(x\) and \(y\) can be represented in the form \(AX = B\) where:

• \(A\) is the coefficient matrix: \( \begin{bmatrix} 1 & 1 \ 0.03 & 0.05 \end{bmatrix} \)
• \(X\) is the variable matrix: \( \begin{bmatrix} x \ y \end{bmatrix} \)
• \(B\) is the constant matrix: \( \begin{bmatrix} 8000 \ 250 \end{bmatrix} \)

This concise representation simplifies the application of Cramer's Rule, where column replacements allow us to compute determinants needed to solve for each variable individually.