Problem 29 - Algebra and Trigonometry

Question

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{c}{2 x-y=-9} \\ {x+2 y=8}\end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Write the System of Equations in Matrix Form

First, identify the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\). The system is:\(\begin{align*} 2x - y &= -9 \ x + 2y &= 8 \end{align*}\). This can be written in matrix form as:\[AX = B\]where \[A = \begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad \text{and} \quad B = \begin{bmatrix} -9 \ 8 \end{bmatrix} \]

Step 2: Compute the Determinant of Matrix A

Compute the determinant of the coefficient matrix \(A\) using the formula for a 2x2 matrix determinant:\[\text{det}(A) = ad - bc\]where \(a, b, c,\) and \(d\) are elements of the matrix \(A = \begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix}\).Thus,\[\text{det}(A) = (2)(2) - (-1)(1) = 4 + 1 = 5\]

Step 3: Compute the Determinant of Modified Matrix A for x (D_x)

Replace the first column of \(A\) with the matrix \(B\) to form a new matrix \(A_x\):\[A_x = \begin{bmatrix} -9 & -1 \ 8 & 2 \end{bmatrix}\]Compute the determinant of \(A_x\):\[\text{det}(A_x) = (-9)(2) - (-1)(8) = -18 + 8 = -10\]

Step 4: Compute the Determinant of Modified Matrix A for y (D_y)

Replace the second column of \(A\) with the matrix \(B\) to form a new matrix \(A_y\):\[A_y = \begin{bmatrix} 2 & -9 \ 1 & 8 \end{bmatrix}\]Compute the determinant of \(A_y\):\[\text{det}(A_y) = (2)(8) - (-9)(1) = 16 + 9 = 25\]

Step 5: Calculate the Value of x Using Cramer's Rule

Using Cramer's Rule, the value of \(x\) is given by:\[x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-10}{5} = -2\]

Step 6: Calculate the Value of y Using Cramer's Rule

Using Cramer's Rule, the value of \(y\) is given by:\[y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{25}{5} = 5\]

Key Concepts

Understanding a System of Equations

A system of equations is simply a set of equations with multiple variables. Solving these allows us to find the exact values for each unknown variable. In our example, we have the system:

• \(2x - y = -9\)
• \(x + 2y = 8\)

Here, these two equations should be considered simultaneously, as they represent two relationships between the variables \(x\) and \(y\). The solution involves finding the specific values of \(x\) and \(y\) that satisfy both equations at once.
Finding solutions to such systems can be approached using different methods in linear algebra, among which Cramer's Rule offers a structured way, especially when dealing with two equations and two unknowns, perfect for a 2x2 matrix! Understanding the system thoroughly is the key first step in using Cramer's Rule.

The Role of Matrix Determinant

The concept of the determinant is crucial in linear algebra, especially when solving systems of equations with Cramer's Rule. The determinant of a matrix provides a scalar value that can be critical for understanding the solutions to a matrix equation.
For a simple 2x2 matrix, the determinant helps determine if the system of equations is solvable and unique. The formula for finding the determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is:\[ \text{det}(A) = ad - bc \]If the determinant of the coefficient matrix \(A\) is zero, the system may not have a unique solution or might even be unsolvable. In our exercise, the determinant was found to be 5, confirming the system has a unique set of solutions. Understanding how to calculate and use determinants is essential when applying Cramer's Rule.

Connecting Linear Algebra with Cramer's Rule

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations, among other things. Cramer's Rule is one of many techniques in linear algebra used to solve linear equations or systems of equations.
By interpreting a system of equations in matrix form \(AX = B\), where \(A\) is the matrix of coefficients, \(X\) is the matrix of variables, and \(B\) is the matrix of constants, we apply operations that are based on the principles of linear algebra.
Cramer's Rule uses the determinants of these matrices to find solutions. It is especially useful for small-sized systems, like our 2x2 system in this exercise, ensuring we can reliably determine the values for variables \(x\) and \(y\). This synergy of concepts allows the application of elegant mathematical techniques like Cramer's Rule.

Using the 2x2 Matrix to Solve Equations

In many cases, when dealing with systems of linear equations, the problem boils down to handling the coefficients effectively. A 2x2 matrix is perfect for organizing these coefficients in our case, where we have two equations and two unknowns.
The matrix form aligns with the equations as follows:

• Matrix \(A\) (coefficients): \(\begin{bmatrix} 2 & -1 \ 1 & 2 \end{bmatrix}\)
• Variable matrix \(X\): \(\begin{bmatrix} x \ y \end{bmatrix}\)
• Constant matrix \(B\): \(\begin{bmatrix} -9 \ 8 \end{bmatrix}\)

This setup allows us to utilize Cramer's Rule. By replacing columns of \(A\) with \(B\), and computing new determinants (e.g., \(A_x\) and \(A_y\)), we solve for \(x\) and \(y\) using:

• \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\)
• \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\)

Working with a 2x2 matrix eases the complexity, allowing a clear path from equations to solutions.