Problem 48 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 5 x-4 y=20 \\ 10 x-8 y=30 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent, with infinitely many solutions.

Step 1: Identify the Coefficient Matrix

The system of equations given is:\[\begin{aligned} 5x - 4y &= 20 \ 10x - 8y &= 30 \end{aligned}\] The coefficient matrix \(A\) for this system is:\[A = \begin{bmatrix} 5 & -4 \ 10 & -8 \end{bmatrix}\]

Step 2: Calculate the Determinant of the Coefficient Matrix

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated as \(ad - bc\).For our matrix \(A\), calculate:\[\text{det}(A) = (5)(-8) - (10)(-4) = -40 + 40 = 0\]Since the determinant is 0, this indicates the matrix is singular.

Step 3: Interpret the Result

When the determinant of the coefficient matrix is zero, it means there is no unique solution for the system of equations.A zero determinant implies the system is either inconsistent or has dependent equations.Since one equation is a multiple of the other (\(10x - 8y = 2 \times (5x - 4y)\)), the equations are dependent.

Key Concepts

Understanding a System of Equations

A system of equations is a collection of two or more equations involving the same set of variables. In this context, the variables are typically represented as \( x \), \( y \), and possibly others, depending on the complexity of the system. The goal when working with systems of equations is to find values for these variables that satisfy all equations in the system simultaneously.In the given exercise, the system of equations can be represented as:

• \( 5x - 4y = 20 \)
• \( 10x - 8y = 30 \)

These equations form a linear system because both are first-degree polynomials. Solving such a system can help us understand relationships between the variables \( x \) and \( y \). There are various methods for solving systems of equations, such as substitution, elimination, and graphing. More advanced methods include using matrix operations, like those seen in Cramer's Rule.

The Role of the Determinant of a Matrix

The determinant of a matrix is a special number that can reveal important properties of the matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \((ad - bc)\). Why is it important? The determinant helps evaluate the solvability of a system of equations.When applying Cramer's Rule to solve a system of equations, the determinant of the coefficient matrix becomes crucial. If the determinant is not zero, the system has a unique solution. However, if the determinant is zero, as is the case in our exercise matrix \(\begin{bmatrix} 5 & -4 \ 10 & -8 \end{bmatrix}\), it suggests no unique solution exists. The determinant gives us insights into whether the equations are dependent or if the system is inconsistent.

Recognizing Dependent Equations

Dependent equations in a system are essentially multiple equations that express the same relationship between variables. They do not provide new information but are rather repeats or multiples of each other.In the provided system, notice how the second equation \(10x - 8y = 30\) is precisely two times the first equation \(5x - 4y = 20\). This indicates dependency because multiplying the entire first equation by 2 will yield the second equation. In mathematical terms, dependent equations are linearly dependent, as they lie on the same line when graphed, and cannot have a unique solution. Identifying such dependencies is crucial because it helps clarify why the system's determinant is zero and confirms the lack of independent solutions.

Identifying Inconsistent Systems

An inconsistent system occurs when there are no solutions that satisfy all equations simultaneously. For instance, if after simplification or transformation, the equations lead to a statement that is always false, like \(0 = 1\), then the system is inconsistent.For the system described in the exercise, one might initially mistake it for being inconsistent due to its singular matrix (determinant equals zero). However, further inspection shows dependence rather than inconsistency. If the equations were inconsistent, they would represent parallel lines with no intersection point, unlike dependent equations which represent the same line.Understanding the distinction between dependent and inconsistent systems is vital when analyzing the solvability and characteristics of a system of equations.