Problem 29 - Precalculus: Mathematical for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{array}{rr} 2 x-y= & -9 \\ x+2 y= & 8 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = -2 \) and \( y = 5 \).

Step 1: Write the system of equations in matrix form

The given system of equations is: \( 2x - y = -9 \) and \( x + 2y = 8 \). In matrix form, this can be written as \( AX = B \) where \[ A = \begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad B = \begin{bmatrix} -9 \ 8 \end{bmatrix} \].

Step 2: Calculate the determinant of the coefficient matrix (det(A))

For the matrix \( A = \begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix} \), the determinant is calculated as: \[ \text{det}(A) = (2 \times 2) - (-1 \times 1) = 4 + 1 = 5 \].

Step 3: Calculate the determinant of matrix Aₓ (det(Aₓ))

Matrix \( Aₓ \) is formed by replacing the first column of \( A \) with \( B \), yielding \( Aₓ = \begin{bmatrix} -9 & -1 \ 8 & 2 \end{bmatrix} \). The determinant is: \[ \text{det}(Aₓ) = (-9 \times 2) - (-1 \times 8) = -18 + 8 = -10 \].

Step 4: Calculate the determinant of matrix A_y (det(A_y))

Matrix \( A_y \) is formed by replacing the second column of \( A \) with \( B \), resulting in \( A_y = \begin{bmatrix} 2 & -9 \ 1 & 8 \end{bmatrix} \). The determinant is: \[ \text{det}(A_y) = (2 \times 8) - (1 \times -9) = 16 + 9 = 25 \].

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

Using Cramer's Rule, the solutions are obtained by:\[ x = \frac{\text{det}(Aₓ)}{\text{det}(A)} = \frac{-10}{5} = -2 \]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{25}{5} = 5 \].

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables that you aim to solve simultaneously. In this case, we have two equations with two unknowns, namely, \(x\) and \(y\). These equations can be described mathematically as follows:

• Equation 1: \(2x - y = -9\)
• Equation 2: \( x + 2y = 8 \)

To find the values of \(x\) and \(y\) that satisfy both equations at the same time, we can use techniques like substitution, elimination, or matrix methods such as Cramer's Rule.
Cramer's Rule is particularly handy when dealing with linear systems that are equal in number to the variables involved. It allows us to find each variable individually using determinants, without the need to simultaneously solve the equations in a more traditional algebraic manner. This method requires the system to have a unique solution, which means the determinant of the coefficient matrix must be non-zero.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It gives us important information about the matrix, such as whether it is invertible. For a 2x2 matrix, the determinant can be found using a simple formula based on its elements.
Consider the matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]The determinant of matrix \( A \) is calculated as:\[\text{det}(A) = ad - bc\]In our example, the coefficient matrix is \[A = \begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix}\]By applying the formula: \[\text{det}(A) = (2 \times 2) - (-1 \times 1) = 4 + 1 = 5\]Since the determinant is non-zero, the system of equations represented can be solved using Cramer's Rule. The determinant essentially acts as a scale factor in transformations represented by the matrix, affirming that the system possesses a unique solution.

Matrix Form of Equations

Transforming a system of equations into a matrix form is a systematic approach allowing us to utilize linear algebra techniques for solving. In matrix notation, we express the system as \(AX = B\), where:

• \(A\) is the coefficient matrix formed by the coefficients of the variables in the equations.
• \(X\) is the column matrix containing the variables, \(x\) and \(y\).
• \(B\) is the constant matrix consisting of the constants from the right-hand side of the equations.

Let's consider our given system:\[\begin{align*}2x - y &= -9 \ x + 2y &= 8\end{align*}\]We can express this in matrix form as:\[A = \begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad B = \begin{bmatrix} -9 \ 8 \end{bmatrix}\]This structure allows us to leverage algebraic operations on matrices, such as finding determinants, to find the solution to the system through methods like Cramer's Rule. By reorganizing the system this way, it becomes easier to apply computational techniques for solving otherwise complex algebraic relationships.