Problem 36 - Precalculus: Functions and Graphs

Question

Use Cramer's rule, whenever applicable, to solve the system. $$\left\\{\begin{array}{rr} 7 x-8 y= & 9 \\ 4 x+3 y= & -10 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = -1 \) and \( y = -2 \).

Step 1: Write the System in Matrix Form

The given system of equations can be represented using matrices. The coefficient matrix, variable matrix, and constant matrix are: \[ A = \begin{pmatrix} 7 & -8 \ 4 & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 9 \ -10 \end{pmatrix} \].So, the system in matrix form is \( AX = B \).

Step 2: Calculate the Determinant of the Coefficient Matrix

To apply Cramer's rule, we first compute the determinant of the coefficient matrix, \( A \). The determinant is calculated as:\[\text{det}(A) = \begin{vmatrix} 7 & -8 \ 4 & 3 \end{vmatrix} = (7 \cdot 3) - (-8 \cdot 4) = 21 + 32 = 53.\] Since \( \text{det}(A) eq 0 \), Cramer's rule is applicable.

Step 3: Calculate Determinants for Variables using Cramer's Rule

Using Cramer's rule, we calculate the determinants needed for \( x \) and \( y \):For \( x \): Replace the first column of the matrix \( A \) with \( B \):\[A_x = \begin{pmatrix} 9 & -8 \ -10 & 3 \end{pmatrix}\]\[\text{det}(A_x) = (9 \cdot 3) - (-8 \cdot -10) = 27 - 80 = -53\]For \( y \): Replace the second column of the matrix \( A \) with \( B \):\[A_y = \begin{pmatrix} 7 & 9 \ 4 & -10 \end{pmatrix}\]\[\text{det}(A_y) = (7 \cdot -10) - (9 \cdot 4) = -70 - 36 = -106\]

Step 4: Solve for the Variables using Cramer's Rule

Now, we use Cramer's rule to solve for \( x \) and \( y \):\( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-53}{53} = -1 \)\( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-106}{53} = -2 \)

Key Concepts

Determinants

Determinants serve as a function that takes a square matrix and produces a single number, often crucial in solving systems of linear equations, particularly in Cramer's rule. For a 2x2 matrix, the determinant is calculated using the formula:\[\text{det}(A) = ad - bc\]where \(A\) is a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \). The determinant provides important information about the matrix, such as if the matrix is invertible or if a system of equations has a unique solution. If a matrix has a determinant of zero, it indicates that the matrix is singular and does not have an inverse, meaning the system of equations may not have a unique solution.

• Non-zero Determinants: Suggest a unique solution.
• Zero Determinants: Indicate either no solutions or infinitely many solutions.

In the context of our exercise, the determinant of the coefficient matrix came out to be 53, which is not zero. This indicates that the system does have a unique set of solutions, making Cramer's rule applicable.

Systems of Equations

Systems of equations consist of multiple equations that share the same set of unknowns. The goal is to find values for these unknowns that satisfy all equations simultaneously. Systems can be represented in different ways, including algebraically, graphically, or in matrix form. In linear algebra, most systems can be encapsulated by matrices and solved through various methods, such as:

• Graphical methods
• Substitution and elimination
• Matrix methods like Gaussian elimination and Cramer's rule

The given problem involves a 2x2 system where the coefficients of the variables are arranged in a matrix, helping to visualize and manipulate the system more elegantly using matrix algebra. Using Cramer's rule, one can solve the system by essentially breaking down the problem into determinant calculations to find the values of the unknowns. In this exercise, a precise set of \(x\) and \(y\) values will satisfy both equations without contradiction, demonstrating its utility for small, manageable systems.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations that can be performed on them. This includes addition, subtraction, multiplication, and the computation of determinants and inverses. When handling systems of equations like the one in this exercise, matrices serve to streamline the computation.

• Matrix Representation: Transforms systems of equations into an easy-to-manage structure.
• Operations: Allow for various manipulations such as multiplication, useful for transforming and solving systems.

Cramer's rule leverages matrix algebra by requiring the computation of determinants from matrices derived by substituting columns. In our case, the coefficient matrix \(A\) was transformed into matrices \(A_x\) and \(A_y\) to compute determinants that directly helped in solving for \(x\) and \(y\). This method showcases how matrices simplify handling multiple equations by transmuting them into a format where determinant calculations become the main tool for finding variable solutions.