Question

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable. \(\left\\{\begin{array}{rr}x-y+2 z= & 5 \\ 3 x+2 y= & 4 \\ -2 x+2 y-4 z= & -10\end{array}\right.\)

Short answer

Not applicable

Step-by-step solution

- Identify the Coefficient Matrix

Identify the coefficient matrix from the system of equations. The coefficient matrix is formed by the coefficients of the variables from each equation.o The given system is:o o o o o o\[\begin{array}{rrr|r}x & -y & 2z & = & 5 \3x & 2y & 0z & = & 4 \-2x & 2y & -4z & = & -10\end{array}\]o The coefficient matrix is:o o o o o o\[\begin{array}{rrr}1 & -1 & 2 \3 & 2 & 0 \-2 & 2 & -4\end{array}\]

- Calculate the Determinant of the Coefficient Matrix

Calculate the determinant, \( \Delta \), of the coefficient matrix to determine if Cramer's Rule is applicable.o o o o o o\[\Delta = \begin{vmatrix} 1 & -1 & 2 \3 & 2 & 0 \-2 & 2 & -4\end{vmatrix}\]o o Evaluate the determinant:o o\[\Delta = 1 \cdot (2 \cdot (-4) - 2 \cdot 0) - (-1) \cdot (3 \cdot (-4) - (-2) \cdot 0) + 2 \cdot (3 \cdot 2 - 2 \cdot (-2))\]o Simplifying, we get:o o\[\Delta = 1 \cdot (-8) - (-1) \cdot (-12) + 2 \cdot (6 + 4)\]o o\[\Delta = -8 - 12 + 20 = 0\]

- Determine Applicability of Cramer's Rule

Since the determinant \( \Delta = 0 \), Cramer's Rule is not applicable for this system of equations.o Write the conclusion.

Key concepts

System of Equations

A system of equations is a set of equations with multiple variables that are solved together. Each equation in the system shares the same set of variables, making them interdependent. For example, in the given system:

\[ \begin{array}{rr} x - y + 2z = 5 \ 3x + 2y = 4 \ -2x + 2y - 4z = -10 \end{array} \]

each equation involves the variables \(x\), \(y\), and \(z\). The goal is to find the values of these variables that satisfy all the equations simultaneously. Systems of equations can be solved using various methods, such as substitution, elimination, or matrix methods like Cramer's Rule.

Determinant

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as:

\[ \text{Det} = ad - bc \].

In the context of a 3x3 matrix, such as our coefficient matrix:

\[ \begin{pmatrix} 1 & -1 & 2 \ 3 & 2 & 0 \ -2 & 2 & -4 \end{pmatrix} \]

the determinant is more complex to calculate. Here is the expanded form for a 3x3 matrix determinant:

\[ \text{Det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Taking the values from our matrix, computing step by step yields:

\[ \text{Det} = 1 \times (2 \times -4 - 2 \times 0) - (-1) \times (3 \times -4 - (-2) \times 0) + 2 \times (3 \times 2 - 2 \times -2) \]

which simplifies to:

\[ \text{Det} = -8 - 12 + 20 \].

This results in a determinant of 0, indicating that Cramer's Rule is not applicable.

Coefficient Matrix

The coefficient matrix is derived from the coefficients of the variables in a system of equations. It serves as a compact representation of the system. For the provided system:

\[ \begin{array}{rr} x - y + 2z = 5 \ 3x + 2y = 4 \ -2x + 2y - 4z = -10 \end{array} \]

we extract the coefficients (ignoring the constants on the right side of the equations), forming:

\[ \begin{pmatrix} 1 & -1 & 2 \ 3 & 2 & 0 \ -2 & 2 & -4 \end{pmatrix} \].

Arranging the coefficients into this matrix format is the first step in applying matrix-based methods, such as Cramer's Rule, to solve the system. The structure of the coefficient matrix directly impacts the results from these methods and influences whether they are applicable.

Linear Algebra

Linear algebra is the branch of mathematics concerning vectors, vector spaces, linear transformations, and systems of linear equations. It provides the theoretical foundation for dealing with linear systems in an abstract way. Key components include:

• Matrices: Rectangular arrays of numbers representing linear transformations or systems of linear equations.
• Vectors: Elements of vector spaces, which can be thought of as points or directions in space.
• Determinants: Scalare values derived from square matrices that provide insights into matrix properties.

Understanding linear algebra is essential for using methods like Cramer's Rule and interpreting their results. Cramer's Rule, which involves determinants of matrices, illustrates practical applications of these concepts by offering a systematic approach to solving systems of linear equations.