Problem 22 - Precalculus

Question

Use Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} &3 x-5 y=7\\\ &6 x-10 y=14 \end{aligned}$$

Step-by-Step Solution

Answer

The system cannot be solved using Cramer's rule as the determinant is zero.

Step 1: Identify Coefficient Matrix

First, identify the coefficient matrix of the system of equations. In the given system: \(3x - 5y = 7\), \(6x - 10y = 14\), the coefficient matrix is \(A = \begin{bmatrix} 3 & -5 \ 6 & -10 \end{bmatrix}\).

Step 2: Calculate Determinant of Coefficient Matrix

Calculate the determinant of matrix \(A\). The determinant \(\text{det}(A)\) is calculated using the formula: \(\text{det}(A) = (3)(-10) - (6)(-5)\). Simplifying gives: \(\text{det}(A) = -30 + 30 = 0\).

Step 3: Analyze the Determinant

Since \(\text{det}(A) = 0\), the coefficient matrix is singular, meaning it cannot be inverted. Therefore, Cramer's rule cannot be used to solve this system because it applies only to non-singular matrices with non-zero determinants.

Key Concepts

Coefficient Matrix

In any system of linear equations, the coefficient matrix plays an essential role. It is a matrix composed of all the coefficients from the variables in the equations. For the exercise provided where you have the system \(3x - 5y = 7\) and \(6x - 10y = 14\), our coefficient matrix \(A\) can be expressed as:

• First row: coefficients from the first equation \([3, -5]\).
• Second row: coefficients from the second equation \([6, -10]\).

Putting these rows into a single matrix, we obtain \(A = \begin{bmatrix} 3 & -5 \ 6 & -10 \end{bmatrix}\).This matrix encodes the structure of the system of equations. It helps us perform further mathematical manipulations like calculating determinants and finding solutions if possible with Cramer's rule.

Determinant Calculation

Calculating the determinant is a critical step when working with Cramer's rule. The determinant helps determine if a unique solution exists. For the matrix \(A = \begin{bmatrix} 3 & -5 \ 6 & -10 \end{bmatrix}\), the determinant \(\text{det}(A)\) is computed by the formula:\[ \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \]In our case, it becomes:

• The product of elements diagonally: \((3)(-10) = -30\).
• The product of the other diagonal: \((6)(-5) = -30\).

Thus, \(\text{det}(A) = -30 + 30 = 0\). A zero determinant indicates that the associated matrix is singular, which is crucial for the next step.

Singular Matrix Analysis

When the determinant of a matrix is zero, the matrix is called singular. This means the matrix does not have an inverse. In the context of Cramer's rule, this is significant because the rule requires the determinant to be non-zero to solve for unique solutions.Observing \(\text{det}(A) = 0\) means our coefficient matrix \(A\) is singular. Hence, the system of equations \(3x - 5y = 7\) and \(6x - 10y = 14\) does not have a unique solution. In practical terms, this often suggests two possibilities:

• The equations in the system might be dependent, meaning one equation is a multiple of the other.
• The equations could potentially represent the same line or plane, indicating that the system has infinitely many solutions or no consistent solutions at all.

Understanding whether a matrix is singular helps prevent misapplication of mathematical methods requiring an inverse, ensuring accuracy in problem-solving.