Problem 31 - College Algebra with Modeling and Visualization

Question

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{rr} 1.7 x-2.5 y= & -0.91 \\ -0.4 x+0.9 y= & 0.423 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( x \approx 0.45 \) and \( y \approx 0.67 \).

Step 1: Identify the coefficients of the system

The given system of equations is: \( 1.7x - 2.5y = -0.91 \) and \( -0.4x + 0.9y = 0.423 \). From these equations, identify the coefficients: \( a_1 = 1.7 \), \( b_1 = -2.5 \), \( c_1 = -0.91 \), \( a_2 = -0.4 \), \( b_2 = 0.9 \), and \( c_2 = 0.423 \).

Step 2: Construct the coefficient matrix and calculate its determinant

Create the coefficient matrix \( A \) using the coefficients of \( x \) and \( y \): \[A = \begin{bmatrix} 1.7 & -2.5 \ -0.4 & 0.9 \end{bmatrix}\]Calculate the determinant of \( A \): \[\text{det}(A) = (1.7)(0.9) - (-0.4)(-2.5) = 1.53 - 1.0 = 0.53\]

Step 3: Form matrices for determinants of numerators for Cramer's Rule

To find \( x \): Replace the first column of \( A \) with the constants from the right side of the equations:\[A_x = \begin{bmatrix} -0.91 & -2.5 \ 0.423 & 0.9 \end{bmatrix}\]To find \( y \): Replace the second column of \( A \) with the constants:\[A_y = \begin{bmatrix} 1.7 & -0.91 \ -0.4 & 0.423 \end{bmatrix}\]

Step 4: Calculate determinants of modified matrices

Calculate \( \text{det}(A_x) \):\[\text{det}(A_x) = (-0.91)(0.9) - (0.423)(-2.5) = -0.819 + 1.0575 = 0.2385\]Calculate \( \text{det}(A_y) \):\[\text{det}(A_y) = (1.7)(0.423) - (-0.4)(-0.91) = 0.7191 - 0.364 = 0.3551\]

Step 5: Apply Cramer's Rule to find the values of x and y

Using Cramer's Rule, solve for \( x \) and \( y \):\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{0.2385}{0.53} \approx 0.45 \]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{0.3551}{0.53} \approx 0.67 \]

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with two or more variables working together. The main objective is to find the values of these variables that satisfy all the equations simultaneously. Linear equations are called so because they graph as straight lines. For example, the system:

• \(1.7x - 2.5y = -0.91\)
• \(-0.4x + 0.9y = 0.423\)

Requires us to solve for \(x\) and \(y\) such that both equations are true at the same time. To tackle these equations efficiently, mathematicians use various methods like substitution, elimination, and Cramer's rule. Cramer's rule provides a clear path to finding solutions if the number of equations equals the number of unknowns and the determinant of the coefficient matrix is non-zero.

Determinant

Determinants play a crucial role in linear algebra, especially when dealing with matrices. The determinant is a special number that can be calculated from a square matrix. It helps determine whether a system of linear equations has a unique solution. If the determinant of the coefficient matrix is non-zero, the system has a unique solution.

• For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( \text{det}(A) = ad - bc \).

In our example, we calculated the determinant of the coefficient matrix:

• \( \text{det}(A) = (1.7)(0.9) - (-0.4)(-2.5) = 1.53 - 1.0 = 0.53 \)

A non-zero determinant (0.53) indicates that the system can be solved using Cramer's rule.

Coefficient Matrix

A coefficient matrix is a concise way to organize the coefficients of the variables from the system of linear equations. In our problem, the coefficient matrix \( A \) includes only the coefficients of \( x \) and \( y \):

• \( A = \begin{bmatrix} 1.7 & -2.5 \ -0.4 & 0.9 \end{bmatrix} \).

This matrix essentially extracts the numerical values in front of the variables, allowing us to focus on how they interact through matrix operations. When using Cramer's rule, we modify this matrix to form other matrices (like \( A_x \) and \( A_y \)) by replacing columns which helps in calculating the values of the variables.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear mappings between these spaces. It is essential for understanding systems of linear equations, matrices, and determinants. With linear algebra, we can leverage matrix notation to simplify and solve complex problems involving several variables.
By setting up the problem as matrices and leveraging operations like finding determinants, we can use techniques like Cramer's rule. This provides straightforward solutions to systems of equations by creating new matrices with determinants that help solve for variable values by simple division. Linear algebra tools are used extensively in fields like computer science, engineering, physics, and any domain that involves modeling and solving multiple interacting linear equations.