Problem 1 - Algebra 2

Question

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{2 x+y=4} \\ {3 x-y=6}\end{array}\right. $$

Step-by-Step Solution

Answer

Using Cramer's Rule, the solution to the system is \(x = 2\) and \(y = 0\).

Step 1 - Write down the coefficient matrix and the constant matrix

Begin by identifying the coefficients of the variables in the system and organize them into a matrix, and then identify the constants on the right-hand side of the equations and organize them into a separate matrix. The coefficient matrix for the given system is \(A = \begin{bmatrix} 2 & 1 \ 3 & -1 \end{bmatrix}\) and the constant matrix is \(B = \begin{bmatrix} 4 \ 6 \end{bmatrix}\).

Step 2 - Calculate the determinant of the coefficient matrix

Find the determinant of the coefficient matrix \(A\). This is done by computing \(\det(A) = (2 \times -1) - (3 \times 1) = -2 - 3 = -5\).

Step 3 - Create matrices for the individual variables

Form new matrices for \(x\) and \(y\) by replacing the corresponding column in matrix \(A\) with the constant matrix \(B\). This gives us \(A_x = \begin{bmatrix} 4 & 1 \ 6 & -1 \end{bmatrix}\) for \(x\), and \(A_y = \begin{bmatrix} 2 & 4 \ 3 & 6 \end{bmatrix}\) for \(y\).

Step 4 - Calculate the determinant of the matrices corresponding to each variable

Find the determinants of these new matrices \(A_x\) and \(A_y\). We get \(\det(A_x) = (4 \times -1) - (6 \times 1) = -4 - 6 = -10\) for \(x\), and \(\det(A_y) = (2 \times 6) - (3 \times 4) = 12 - 12 = 0\) for \(y\).

Step 5 - Solve for the variables using Cramer's Rule

Use Cramer's Rule to solve for \(x\) and \(y\). According to Cramer's Rule, \(x = \frac{\det(A_x)}{\det(A)}\) and \(y = \frac{\det(A_y)}{\det(A)}\). Plugging in our determinant values, we have \(x = \frac{-10}{-5} = 2\) and \(y = \frac{0}{-5} = 0\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix. It is of paramount importance in linear algebra, as it is used to determine if a system of linear equations has a unique solution, and it's a core component of Cramer's Rule.

For a 2x2 matrix, the determinant is calculated using the formula: \[ \text{If } A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \text{ then } \det(A) = ad - bc \.\] This formula is what we used in the solution of the given exercise to calculate the determinant of the coefficient matrix. A nonzero determinant signifies a unique solution to the system of equations; if the determinant is zero, the system may have infinitely many solutions or no solution at all. In the context of our exercise, the determinant of matrix A was -5, indicating that a unique solution exists for our system of equations.

System of Linear Equations

A system of linear equations consists of two or more equations made up of two or more variables that are to be solved simultaneously. In the exercise, we have two equations with two variables, x and y:
\[2x + y = 4\]
\[3x - y = 6\]

Such a system can be solved using various methods, including graphing, substitution, elimination, or more advanced techniques like matrix operations or applying Cramer's Rule. The goal is to find the values of x and y that satisfy both equations at the same time. In our exercise, we used Cramer's Rule because it provides a straightforward procedure for finding the solutions, especially when there is a unique solution, which is the case when the determinant of the coefficient matrix is non-zero.

Coefficient Matrix

In a system of linear equations, each variable is associated with a coefficient, forming what we call the coefficient matrix. The coefficient matrix is a key concept when dealing with systems of equations, as it organizes these coefficients in a structured form and prepares them for matrix operations.

In the example given, the coefficient matrix is:\[A = \begin{bmatrix} 2 & 1 \ 3 & -1 \end{bmatrix}\]
This matrix includes the coefficients of x and y from both equations of our system. By using matrix notation, we can separate the structural aspects of the system (the coefficients) from the specific values on the right-hand side of the equations. This separation is what enables methods like Cramer's Rule to work, as it relies on the manipulation of the coefficient matrix and its determinants to find the solution to the system of equations.