Question

Solve the following systems of three linear (c) \(i_{1}+i_{2}-i_{3}=1\) equations using Cramer's rule: (a) \(2 x-y-2 z=1\) \(x+y+3 z=6\) \(-2 x+3 z=-1\) (b) \(-2 x+3 z=-1\) \(\alpha+3 \beta+\gamma=2\) \(3 \alpha+2 \gamma=3\) \(v_{1}+\frac{1}{2} v_{2}-v_{3}=0.7\) \(-3 v_{1}+2 v_{3}=-3.1\) \(\theta_{1}+\theta_{2}+4 \theta_{3}=0\) \(2 \theta_{1}-\theta_{2}+\theta_{3}=5.4\) \(\theta_{1}+2 \theta_{2}-\theta_{3}=9.2\)

Short answer

Given the following solution to a system of linear equations using Cramer's rule: \[x=2, y=3\frac{4}{11}, z=1\frac{2}{11}\] Explain the geometrical interpretation of Cramer's rule and how it is used in solving this system of linear equations.

Step-by-step solution

Identify the system of linear equations

We are given the following system of linear equations for system (a): $$2x - y - 2z = 1$$ $$x + y + 3z = 6$$ $$-2x + 3z = -1$$

Write down the coefficient matrix and its determinant

Create a coefficient matrix \(A\) from the given system of linear equations and compute its determinant: $$ A = \begin{bmatrix} 2 & -1 & -2 \\ 1 & 1 & 3 \\ -2 & 0 & 3 \end{bmatrix} $$ Now, find the determinant of matrix \(A\) (denoted as \(|A|\)): $$ |A| = 2\begin{vmatrix}1 & 3\\0 & 3\end{vmatrix} - (-1)\begin{vmatrix}1 & 3\\-2 & 3\end{vmatrix} -2\begin{vmatrix}1 & 1\\-2 & 0\end{vmatrix} $$ Now, calculate the determinants inside: $$ |A| = 2(1\cdot 3 - 0) -(-1)(1\cdot 3 - (-2)\cdot 3) - 2(1\cdot 0 - (-2)\cdot1) $$ $$ |A| = 6 + 9 - 4 = 11 $$

Find the determinants with variables

Replace the first column of matrix \(A\) with the constants from the equations, then find its determinant. Repeat this process for the second and third columns: $$ A_{x} = \begin{bmatrix} 1 & -1 & -2 \\ 6 & 1 & 3 \\ -1 & 0 & 3 \end{bmatrix} $$ $$ |A_{x}| = 1\begin{vmatrix}1 & 3\\0 & 3\end{vmatrix} - (-1)\begin{vmatrix}6 & 3\\-1 & 3\end{vmatrix} -2\begin{vmatrix}6 & 1\\-1 & 0\end{vmatrix} $$ $$ |A_{x}| = 3 + 21 - 2 = 22 $$ $$ A_{y} = \begin{bmatrix} 2 & 1 & -2 \\ 1 & 6 & 3 \\ -2 & -1 & 3 \end{bmatrix} $$ $$ |A_{y}| = 2\begin{vmatrix}6 & 3\\-1 & 3\end{vmatrix} - 1\begin{vmatrix}1 & 3\\-2 & 3\end{vmatrix} -2\begin{vmatrix}1 & 6\\-2 & -1\end{vmatrix} $$ $$ |A_{y}| = 30 - 1 + 8 = 37 $$ $$ A_{z} = \begin{bmatrix} 2 & -1 & 1 \\ 1 & 1 & 6 \\ -2 & 0 & -1 \end{bmatrix} $$ $$ |A_{z}| = 2\begin{vmatrix}1 & 6\\0 & -1\end{vmatrix} - (-1)\begin{vmatrix}1 & 6\\-2 & -1\end{vmatrix} - 1\begin{vmatrix}1 & 1\\-2 & 0\end{vmatrix} $$ $$ |A_{z}| = -2 + 13 + 2 = 13 $$

Compute the values of x, y, and z

Divide the determinants \(|A_x|\), \(|A_y|\), and \(|A_z|\) by \(|A|\) to find the values of \(x\), \(y\), and \(z\): $$ x = \frac{|A_x|}{|A|} = \frac{22}{11} = 2 $$ $$ y = \frac{|A_y|}{|A|} = \frac{37}{11} = 3\frac{4}{11} $$ $$ z = \frac{|A_z|}{|A|} = \frac{13}{11} = 1\frac{2}{11} $$ So, the solution for system (a) is: $$ x=2, y=3\frac{4}{11}, z=1\frac{2}{11} $$

Key concepts

Understanding Systems of Linear Equations

A system of linear equations consists of two or more linear equations that have common variables. A linear equation is an equation of the form: \(ax + by + cz + \ldots = d\). Here, \(x, y, z,\) and so on are variables, and \(a, b, c,\) and \(d\) are constants. In most cases, solving a system of linear equations means finding the values of these variables that make all equations true simultaneously. Systems of equations can be represented in matrix form, which helps to apply different methods to find solutions. One such method is Cramer's Rule, which can be used when the determinant of the coefficient matrix is not zero. This method offers a structured approach to finding the solution by isolating each variable using determinants.

Determinant and Its Role in Cramer's Rule

A determinant is a scalar value that can be computed from the elements of a square matrix. This value can provide critical insights into the properties of the matrix. For instance, the determinant gives us information about the matrix's invertibility. If the determinant is non-zero, the matrix is invertible, indicating a unique solution exists for the system of equations when using Cramer's Rule. Conversely, a zero determinant means that the matrix is not invertible, and the system may either have infinitely many solutions or none. Computing the determinant involves a specific process of multiplying and adding the elements of the matrix, significantly affected by the matrix's size. In our solution, the determinant of the coefficient matrix was calculated to be non-zero (11), enabling the use of Cramer's Rule.

Matrix Operations in Solving Systems

Matrix operations include actions like addition, subtraction, and finding determinants, essential for solving systems with methods like Cramer's Rule. Begin by arranging the coefficients of the variables into a square matrix. The original example uses a 3x3 matrix derived from the coefficients of the given equations. Each column corresponds to the coefficients of a particular variable. To apply Cramer's Rule, replace each column of the coefficient matrix one at a time with the constants from the corresponding equations. Compute the determinant for each of these modified matrices. The solutions to the variables \(x, y,\) and \(z\) are derived by dividing these determinants by the original matrix's determinant. Therefore, understanding and applying matrix operations is crucial to calculating solutions efficiently.