Problem 50 - 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. $$ \left\\{\begin{array}{l} 16 x-8 y=32 \\ x-2=\frac{y}{2} \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Write the system in standard form

First, bring the system of equations into the standard form \(Ax + By = C\). The given system is: \(\begin{cases} 16x - 8y = 32 \ x - 2 = \frac{y}{2} \end{cases}\). Rewrite the second equation: \(x - \frac{y}{2} = 2\), which becomes \(2x - y = 4\) when multiplied by 2.

Step 2: Define the coefficients

Define the coefficients for each equation. The system is: \(\begin{cases} 16x - 8y = 32 \ 2x - y = 4 \end{cases}\). So we have \(A = \begin{bmatrix} 16 & -8 \ 2 & -1 \end{bmatrix}\) and \(C = \begin{bmatrix} 32 \ 4 \end{bmatrix}\).

Step 3: Calculate the determinant of the coefficient matrix

Calculate the determinant of the matrix \(A\). Det\(A\) is found using the formula for a 2x2 matrix: \(Det(A) = (16)(-1) - (2)(-8) = -16 + 16 = 0\).

Step 4: Conclusion about the system

Since the determinant of the coefficient matrix \(Det(A) = 0\), Cramer's Rule cannot be applied. This indicates that the system of equations is either dependent or inconsistent. To determine which, check for consistency and see if one equation is a multiple of the other.

Step 5: Check for dependency

The second equation \(2x - y = 4\) can be rewritten as \(16x - 8y = 32\) by multiplying by 8, which is identical to the first equation. This means the system is dependent and has infinitely many solutions.

Key Concepts

Determinants

Determinants are a key concept in solving systems of linear equations, especially when using methods like Cramer's Rule. They are calculated from square matrices and help to determine the properties of the system.
For a 2x2 matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the determinant is calculated as:\[Det(A) = ad - bc.\]

• If the determinant is not zero, it indicates that the matrix is invertible, and the system of equations has a unique solution.
• If the determinant is zero, like in our step-by-step solution, it suggests that Cramer's Rule can't be used directly. Here, the system may be either dependent or inconsistent.

These results are crucial in determining the path to solve the system or assess its nature.

Linear Equations

Linear equations are mathematical statements of equality involving linear expressions. Each equation in a system represents a line in two-dimensional space.
For example, the given equations:\[16x - 8y = 32\]and \[2x - y = 4\]express relationships between the variables \(x\) and \(y\).

• Each equation can be manipulated and transformed into standard form.
• The goal is often to find values of \(x\) and \(y\) that satisfy all equations simultaneously.

Linear equations form the basis of systems in linear algebra, providing simple forms for theoretical and practical solutions.

Dependent System

A dependent system of linear equations occurs when the equations essentially describe the same line or plane. This means they are scalar multiples of each other.
In our exercise, the system of equations was found to be dependent because the second equation:\[2x - y = 4\]can be transformed into:\[16x - 8y = 32\]by multiplying through by 8. This shows that the two equations are not independent, and they represent the same line.

• Dependent systems indicate that there are infinitely many solutions.
• Graphically, this means the lines or planes overlap completely.

Understanding dependency is vital for evaluating the nature of solutions we can have in a system.

Inconsistent System

An inconsistent system of linear equations has no solutions, which occurs when the equations represent parallel lines that never intersect.
It often arises when the determinant of the coefficient matrix is zero but cannot be resolved by simple scalar multiplication, as opposed to a dependent system.
However, in our specific exercise, the system was not inconsistent because the second equation could be derived by multiplying the first equation. Thus, the lines were coincident, not parallel.

• Inconsistent systems are characterized by contradictory equations with no point of intersection.
• They result in impossible situations, like a statement that 0 equals a non-zero value.

Identifying inconsistencies helps confirm the lack of any possible solution.