Problem 49 - 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. $$ \left\\{\begin{array}{l} \frac{5}{6} x=2-y \\ 10 x+12 y=24 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Rewrite the System of Equations

First, let's rewrite the system of equations in a standard linear form (Ax + By = C). The given equations are: 1. \( \frac{5}{6}x = 2 - y \) 2. \( 10x + 12y = 24 \). For equation 1, rearrange terms to put it in the form of \( Ax + By = C \): Multiply through by 6 to clear the fraction: \[ 5x = 12 - 6y \] Rearrange to: \[ 5x + 6y = 12 \] Now, the system of equations is: 1. \( 5x + 6y = 12 \) 2. \( 10x + 12y = 24 \).

Step 2: Write the Coefficient Matrix and Determinants

The coefficient matrix \( A \) for the system is: \[ A = \begin{bmatrix} 5 & 6 \ 10 & 12 \end{bmatrix} \]Cramer's rule requires the calculation of the determinant of matrix \( A \), \( \text{det}(A) \):\[ \text{det}(A) = (5)(12) - (6)(10) = 60 - 60 = 0 \]

Step 3: Interpret the Determinant Result

Since \( \text{det}(A) = 0 \), the coefficient matrix is singular, which means the system does not have a unique solution. This can imply that the system is either dependent (infinitely many solutions) or inconsistent (no solution).

Step 4: Check for Consistency or Dependency

To further analyze, compare the two equations:Multiply the first equation by 2: \[ 10x + 12y = 24 \] This is identical to the second equation. Hence, the system is dependent, meaning the equations are scalar multiples of each other and there are infinitely many solutions.

Key Concepts

Systems of Equations

In algebra, a system of equations consists of two or more equations that share the same set of variables. Typically, these are written in the form \( Ax + By = C \). Here, the given system includes two linear equations, which means they graph as straight lines in a coordinate plane.

When working with a system of two equations, we're often solving for two variables – in this case, \( x \) and \( y \). The goal is to find values that satisfy both equations simultaneously:

• Equation 1: \( \frac{5}{6}x = 2 - y \)
• Equation 2: \( 10x + 12y = 24 \)

The key is to rewrite these equations in a linear form, as done in the step-by-step solution, where they become:

• \( 5x + 6y = 12 \)
• \( 10x + 12y = 24 \)

Understanding how to manipulate these equations into a solvable form is a fundamental skill when dealing with algebra. The solution methods can range from graphing, substitution, elimination, to more advanced techniques like using matrices and determinants.

Determinants

When solving a system of linear equations using Cramer's Rule, calculating the determinant of a matrix is essential. A determinant is a special number that can be calculated from a square matrix and provides critical information about the matrix's properties.

For a 2x2 coefficient matrix as in our exercise, \( A = \begin{bmatrix} 5 & 6 \ 10 & 12 \end{bmatrix} \), the determinant \( \text{det}(A) \) is calculated as follows:

• Multiply the diagonals: \( (5 \times 12) = 60 \)
• Subtract the other diagonal's multiplication: \( (6 \times 10) = 60 \)
• Find \( \text{det}(A) = 60 - 60 = 0 \)

The determinant being zero means that the system's matrix is singular, which implies there isn't a unique solution. This can either indicate that the system has no solutions, called inconsistent, or infinitely many solutions, termed dependent. Understanding and calculating the determinant helps to determine the system's nature.

Dependent Equations

Dependent equations are a set of equations where one equation is essentially a multiple of another. This implies that they don't provide unique pieces of information, resulting in infinitely many solutions.

In the provided exercise, after rewriting and manipulating the equations, it becomes clear that one equation is a scaled version of the other. By multiplying Equation 1 by two:

• \( 10x + 12y = 24 \)

You produce exactly Equation 2. This confirms the equations are dependent.

When dealing with dependent systems, any set of values for \( x \) and \( y \) that satisfies one equation will automatically satisfy the other. Therefore, in practical terms, understanding this allows us to recognize that solving the system analytically might not be necessary, as the system translates to the same line on a graph, representing countless solutions.