Problem 27 - College Algebra with Modeling and Visualization

Question

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{r} -2 x+3 y=8 \\ 4 x-5 y=3 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 24.5\) and \(y = 19\).

Step 1: Calculate the Determinant of the Coefficient Matrix

The coefficient matrix for the system is \( \begin{bmatrix} -2 & 3 \ 4 & -5 \end{bmatrix} \). Calculate its determinant using the formula: \( \text{det}(A) = ad - bc \). So, \( \text{det}(A) = (-2)(-5) - (3)(4) = 10 - 12 = -2 \).

Step 2: Calculate the Determinant for \(x\)

Replace the first column of the coefficient matrix with the constants from the equations to form the matrix \( \begin{bmatrix} 8 & 3 \ 3 & -5 \end{bmatrix} \). Calculate its determinant: \( \text{det}(A_x) = 8(-5) - 3(3) = -40 - 9 = -49 \).

Step 3: Calculate the Determinant for \(y\)

Replace the second column of the coefficient matrix with the constants from the equations to form the matrix \( \begin{bmatrix} -2 & 8 \ 4 & 3 \end{bmatrix} \). Calculate its determinant: \( \text{det}(A_y) = (-2)(3) - (8)(4) = -6 - 32 = -38 \).

Step 4: Solve for \(x\) using Cramer's Rule

Use Cramer's rule to find \(x\): \( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-49}{-2} = 24.5 \).

Step 5: Solve for \(y\) using Cramer's Rule

Use Cramer's rule to find \(y\): \( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-38}{-2} = 19 \).

Key Concepts

System of Linear Equations

Understanding systems of linear equations is crucial when diving into concepts like Cramer's Rule. These systems consist of multiple linear equations with variables that we need to solve for. For example, the system \[\begin{array}{r}-2 x+3 y=8 \4 x-5 y=3\end{array}\]has two equations with two variables, \(x\) and \(y\).
This means we're seeking values of \(x\) and \(y\) that satisfy both equations simultaneously. Solving such systems can reveal insights into relationships between different quantities, often expressed linearly.
To assess a solution, it should make all the original equations true when substituted back. Techniques like substitution, elimination, and Cramer's Rule help us systematically solve these mathematical puzzles.

Determinant

The concept of a determinant is central to understanding matrix-related solutions like Cramer's Rule. Determinants are special numbers calculated from a square matrix, providing crucial insights into mathematical properties.
For the coefficient matrix \(\begin{bmatrix} -2 & 3 \ 4 & -5 \end{bmatrix}\), the determinant helps determine whether the system has a unique solution.To find the determinant for a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), use the formula:

• \(\text{det}(A) = ad - bc\)

The determinant helps assess the matrix's invertibility, indicating if processes like Cramer's Rule can be used. A non-zero determinant, like \(-2\) in this case, signifies a unique solution exists.

Coefficient Matrix

The coefficient matrix is a simple but powerful representation in linear equations, showcasing the coefficients of the variables.
For our given system, the coefficient matrix is:\[\begin{bmatrix} -2 & 3 \ 4 & -5 \end{bmatrix}\]This matrix highlights the linkage and interactions between variables \(x\) and \(y\).
Each element of the matrix corresponds to coefficients from the equations, assisting in the organization and simplification of calculations.A coefficient matrix is critical for applying techniques like Cramer's Rule and other matrix-based methods to solve linear equations. It forms the foundation upon which we replace columns or calculate determinants, as in our steps towards a solution.

Algebra

Algebra serves as the backbone of solving and manipulating systems of linear equations. It involves the study of symbols and the rules for manipulating these symbols to express general mathematical relationships and solve equations.
Using algebraic methods, we make sense of linear systems like our example, allowing us to apply structured strategies such as:

• Substitution
• Elimination
• Matrix manipulation, including Cramer's Rule

Algebra breaks down complex problems into manageable pieces. It allows us to perform operations that simplify and solve equations, translating real-world problems into mathematical forms that provide clear answers.