Problem 26 - Algebra for College Students

Question

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}x-2 y+z=1 \\ 3 x+y-z=2 \\ 2 x-4 y+2 z=-1\end{array}\right)\)

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions.

Step 1: Identify the System of Equations

The given system of equations can be written as: \( x - 2y + z = 1 \), \( 3x + y - z = 2 \), and \( 2x - 4y + 2z = -1 \). We will use Cramer's rule to solve it.

Step 2: Construct the Coefficient Matrix

The coefficient matrix \( A \) associated with these equations is: \[A = \begin{pmatrix} 1 & -2 & 1 \ 3 & 1 & -1 \ 2 & -4 & 2 \end{pmatrix}\]

Step 3: Determine the Determinant of the Coefficient Matrix

Calculate the determinant of \( A \):\[\text{det}(A) = \begin{vmatrix} 1 & -2 & 1 \ 3 & 1 & -1 \ 2 & -4 & 2 \end{vmatrix}\] Using cofactor expansion along the first row gives:\(\text{det}(A) = 1 \cdot ((1)(2) - (-1)(-4)) + 2 \cdot (3 \cdot 2 - (-1)(2)) + 1 \cdot (3(-4) - 1(2)) = 0\).

Step 4: Interpret the Determinant Result

The determinant of the coefficient matrix \( A \) is zero, indicating that the system of equations is either dependent or inconsistent.

Step 5: Conclusion Based on Zero Determinant

Since the determinant is zero, the system of equations is dependent, meaning there are infinitely many solutions. The coefficients of the third equation are multiples of the first, confirming dependency.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share the same set of unknowns. This concept is fundamental in mathematics, especially when dealing with variables such as \(x\), \(y\), and \(z\), which appear in each equation. Each equation describes a specific condition that the variables must satisfy. In our example, we are given three equations: \(x - 2y + z = 1\), \(3x + y - z = 2\), and \(2x - 4y + 2z = -1\).

The goal is to find values for \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. When solving such a system, you can express these equations in matrix form, which helps in applying various methods like Cramer's Rule for simplicity. By using a matrix, you link the equations systematically, making calculations more streamlined.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. In the context of solving systems of equations using Cramer's Rule, determining whether the system has a unique solution depends on the determinant of the coefficient matrix. For a given matrix \(A\), the determinant is represented by \(\text{det}(A)\).

The determinant can often tell us about the properties of the system:

• If \(\text{det}(A) eq 0\), the matrix has an inverse, and the system of equations has a unique solution.
• If \(\text{det}(A) = 0\), the system might be dependent (having infinitely many solutions) or inconsistent (having no solutions at all).

In our example, we computed \(\text{det}(A) = 0\) using the cofactor expansion method along the first row.

This result means that the equations are dependent as it suggests linear dependence among the equations, hence resulting in infinitely many solutions.

Dependent Equations

Dependent equations in a system refer to the situation where one equation can be derived from another (or others) through some algebraic operations, making them not independent. In our context, the dependence of equations leads to scenarios with infinitely many solutions.

In the given problem, the coefficients of the third equation \(2x - 4y + 2z = -1\) can be obtained by multiplying the first equation \(x - 2y + z = 1\) by \(-1\). This observation confirms the dependence between the equations. Such dependency implies that all points that satisfy one set of conditions automatically satisfy the others, due to their inherent interrelationship.

Hence, when faced with dependent equations, rather than seeking a single solution, understand that the set of solutions includes infinitely many points along a line or plane in the multi-dimensional space.