Question

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable" \(\left\\{\begin{array}{r}2 x-y=-1 \\ x+\frac{1}{2} y=\frac{3}{2}\end{array}\right.\)

Short answer

x = 0.5, y = 2

Step-by-step solution

Identify the system of equations

Given the system of equations: \[\begin{cases} 2x - y = -1 \ x + \frac{1}{2}y = \frac{3}{2} \end{cases}\]

Write the coefficient matrix

The coefficient matrix for the system is: \[ A = \begin{pmatrix} 2 & -1 \ 1 & \frac{1}{2} \end{pmatrix} \]

Calculate the determinant of the coefficient matrix

The determinant of matrix \( A \) is calculated as follows: \[ \text{det}(A) = \begin{vmatrix} 2 & -1 \ 1 & \frac{1}{2} \end{vmatrix} = (2 \cdot \frac{1}{2}) - (-1 \cdot 1) = 1 + 1 = 2 \]

Check if the determinant is zero

Since \( \text{det}(A) eq 0 \), Cramer's Rule is applicable.

Write the matrix for the numerators

The matrix for the numerators are formed by replacing the column of the unknown variable with the constants: \[ A_x = \begin{pmatrix} -1 & -1 \ \frac{3}{2} & \frac{1}{2} \end{pmatrix} \] \[ A_y = \begin{pmatrix} 2 & -1 \ 1 & \frac{3}{2} \end{pmatrix} \]

Calculate the determinants of these new matrices

Calculate \( \text{det}(A_x) \) and \( \text{det}(A_y) \): \[ \text{det}(A_x) = \begin{vmatrix} -1 & -1 \ \frac{3}{2} & \frac{1}{2} \end{vmatrix} = (-1 \cdot \frac{1}{2}) - (-1 \cdot \frac{3}{2}) = -\frac{1}{2} + \frac{3}{2} = 1 \] \[ \text{det}(A_y) = \begin{vmatrix} 2 & -1 \ 1 & \frac{3}{2} \end{vmatrix} = (2 \cdot \frac{3}{2}) - (-1 \cdot 1) = 3 + 1 = 4 \]

Use Cramer's Rule to find the variables

According to Cramer's Rule, the solution for \( x \) and \( y \) is given by: \[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{1}{2} \] \[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{4}{2} = 2 \]

Key concepts

System of Equations

A system of equations is a set of two or more equations with the same set of unknowns. In this exercise, the system is:
\[ 2x - y = -1 \ x + \frac{1}{2}y = \frac{3}{2} \]
Each equation expresses a relationship between the variables \( x \) and \( y \). The goal is to find the values of these variables that satisfy both equations simultaneously.
Cramer's Rule is a method used to solve such systems, provided certain conditions are met, notably the non-zero determinant of the coefficient matrix.
Using Cramer's Rule involves several steps: constructing the coefficient matrix, calculating its determinant, and then forming new matrices to solve for each variable.

Determinant

The determinant of a matrix is a special number that can be calculated from its elements and plays a key role in matrix algebra.
For a 2x2 matrix, the determinant is given by:
\[ \text{det} \left( \begin{matrix} a & b \ c & d \end{matrix} \right) = (a \cdot d) - (b \cdot c) \]
In our problem, we have the coefficient matrix:
\[ A = \begin{pmatrix} 2 & -1 \ 1 & \frac{1}{2} \end{pmatrix} \]
The determinant is calculated as:
\[ \text{det}(A) = (2 \cdot \frac{1}{2}) - (-1 \cdot 1) = 1 + 1 = 2 \]
Since the determinant is not zero, it confirms that we can use Cramer's Rule to solve the system.

Matrix Algebra

Matrix algebra involves various operations like addition, subtraction, multiplication, and finding determinants.
These operations are essential for solving systems of equations using methods like Cramer's Rule.
When we use matrices to represent systems of equations, the coefficients of the variables form a matrix. For instance, the coefficient matrix for our exercise is:
\[ A = \begin{pmatrix} 2 & -1 \ 1 & \frac{1}{2} \end{pmatrix} \]
To solve for variables using matrix algebra, we also form matrices for numerators by replacing columns in the original matrix with the constants from the equations. We calculate determinants for these new matrices and then use them to find the values of the variables.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces.
It provides the tools and methods for solving systems of linear equations, like the one we are given.
Cramer's Rule is derived from linear algebra concepts. It provides a direct way to solve systems of linear equations where the number of equations equals the number of unknowns, and the coefficient matrix has a non-zero determinant.
In applying Cramer's Rule, we use determinants and matrix operations to find unique solutions to the system of equations. With our exercise, we demonstrated how to calculate determinants and use them according to the rule to find the values of \( x \) and \( y \).