Problem 67 - 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. $$ \left\\{\begin{array}{l} 2 x+3 y=0 \\ 4 x-6 y=-4 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has a unique solution: \(x = -\frac{1}{2}\), \(y = \frac{1}{3}\).

Step 1: Determine the Coefficient Matrix

Identify the coefficients from the given system of linear equations. The coefficient matrix for the system is derived from the left-hand side of the equations: \(\begin{bmatrix} 2 & 3 \ 4 & -6 \\end{bmatrix}\).

Step 2: Calculate the Determinant of the Coefficient Matrix

Calculate the determinant of the matrix: \(\det(C) = \begin{vmatrix} 2 & 3 \ 4 & -6 \end{vmatrix} = 2(-6) - 3(4) = -12 - 12 = -24\).

Step 3: Substitute for x using Cramer's Rule

Construct the matrix for x by replacing the first column of the coefficient matrix with the constants:\(\begin{bmatrix} 0 & 3 \ -4 & -6 \\end{bmatrix}\). Calculate the determinant: \(\det(C_x) = \begin{vmatrix} 0 & 3 \ -4 & -6 \end{vmatrix} = 0(-6) - 3(-4) = 12\). Use Cramer's rule for x: \(x = \frac{\det(C_x)}{\det(C)} = \frac{12}{-24} = -\frac{1}{2}\).

Step 4: Substitute for y using Cramer's Rule

Construct the matrix for y by replacing the second column of the coefficient matrix with the constants:\(\begin{bmatrix} 2 & 0 \ 4 & -4 \\end{bmatrix}\). Calculate the determinant: \(\det(C_y) = \begin{vmatrix} 2 & 0 \ 4 & -4 \end{vmatrix} = 2(-4) - 0 = -8\). Use Cramer's rule for y: \(y = \frac{\det(C_y)}{\det(C)} = \frac{-8}{-24} = \frac{1}{3}\).

Step 5: Conclusion

Since the determinant of the coefficient matrix is non-zero, the system is consistent and independent. The solution to the system using Cramer's rule is \(x = -\frac{1}{2}\) and \(y = \frac{1}{3}\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix. It plays a crucial role in solving systems of equations, especially when using methods like Cramer's Rule. In simpler terms, a determinant provides information about the matrix's properties. For example, if a matrix has a non-zero determinant, it means that the matrix has an inverse, and the related system of equations has a unique solution. On the other hand, if the determinant is zero, the system may be dependent or inconsistent.

To calculate the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), use the formula:

• \( det = ad - bc \)

In the original problem, the coefficient matrix was \(\begin{bmatrix} 2 & 3 \ 4 & -6 \end{bmatrix}\). Calculating its determinant involves simple multiplication and subtraction: \(2(-6) - 3(4) = -12 - 12 = -24\). Since this result is non-zero, it assures us that the corresponding system is consistent and has a unique solution. Calculating the determinant forms the initial and essential step in applying Cramer's Rule.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. The goal of solving a system of equations is to find the values of the variables that satisfy all equations simultaneously. These systems can be classified based on the number of solutions. They can be:

• Consistent and Independent: There is exactly one solution.
• Consistent and Dependent: There are infinitely many solutions.
• Inconsistent: There are no solutions that satisfy all equations together.

The original exercise illustrated a system of equations:
\[\begin{align*}2x + 3y &= 0 \4x - 6y &= -4\end{align*}\]By observing the equations, we understand they form a linear system and can be solved using algebraic methods. However, to find solutions more efficiently, techniques like Cramer's Rule can be applied when the conditions are suitable. In this case, the conditions were met since the determinant was non-zero, indicating the system has a unique solution.

Consistent and Independent Systems

Consistent and independent systems are systems of equations that have exactly one solution, meaning that the equations meet at one unique point on the graph. This type of system is characterized by having a non-zero determinant in its coefficient matrix when using Cramer's Rule.
When a system is consistent, it means there is at least one set of values for the unknowns that satisfies every equation in the system. When these solutions are independent, it underscores that solving the system leads to a unique answer with no repetition or dependency between the equations.
In the exercise provided, the solution to the system was found using Cramer's Rule, confirming that it is consistent and independent since:

• The determinant of the coefficient matrix was -24, not zero.
• The calculations resulted in unique values for \(x\) and \(y\) or exactly one pair solution \(x = -\frac{1}{2}\), \(y = \frac{1}{3}\).

These elements together ensure that the system is a clear example of being both consistent and independent, offering a singular solution set.