Problem 38 - Algebra and Trigonometry

Question

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{array}{l}{10 x-17 y=21} \\ {20 x-31 y=39}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{2}{5} \) and \( y = -1 \).

Step 1: Identify the Coefficient Matrix

For the given system of equations, identify the coefficient matrix \( A \). The equations are: \( 10x - 17y = 21 \) and \( 20x - 31y = 39 \). Hence, the coefficient matrix is \(A = \begin{pmatrix} 10 & -17 \ 20 & -31 \end{pmatrix}.\)

Step 2: Compute the Determinant of the Coefficient Matrix

Use the formula for the determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \):\[\text{det}(A) = ad - bc\]Substituting the values, we have:\[\text{det}(A) = (10)(-31) - (-17)(20) = -310 + 340 = 30.\]

Step 3: Formulate the Matrices with Replaced Columns and Compute Determinants

First, replace the first column of \( A \) with the constants to find \( A_x \):\[A_x = \begin{pmatrix} 21 & -17 \ 39 & -31 \end{pmatrix}\]Compute \( \text{det}(A_x) \):\[\text{det}(A_x) = (21)(-31) - (-17)(39) = -651 + 663 = 12.\]Next, replace the second column of \( A \) with the constants to find \( A_y \):\[A_y = \begin{pmatrix} 10 & 21 \ 20 & 39 \end{pmatrix}\]Compute \( \text{det}(A_y) \):\[\text{det}(A_y) = (10)(39) - (21)(20) = 390 - 420 = -30.\]

Step 4: Apply Cramer's Rule to Solve for Variables

According to Cramer's Rule, the solutions for \( x \) and \( y \) are given by:\[x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}\]Substituting the determinants:\[x = \frac{12}{30} = \frac{2}{5},\]\[y = \frac{-30}{30} = -1.\]

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number calculated from a square matrix. In the context of solving systems of linear equations using Cramer's Rule, the determinant plays a crucial role. When you have a matrix:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]the determinant is found using the formula:\[\text{det}(A) = ad - bc\]This formula gives a scalar value that can indicate various properties of the matrix, such as whether it is invertible. For example, in the exercise, the coefficient matrix had a determinant of 30. Since this is not zero, Cramer's Rule is applicable, meaning the system of equations has a unique solution. The determinant essentially "scales" how much the vector space defined by the matrix will be transformed, which has computational importance when solving linear equations.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. It can visually be thought of as several lines in a plane, as in 2D, where each equation represents a line. The solution to the system is the point or points where these lines intersect. In our exercise, the system:

• 10x - 17y = 21
• 20x - 31y = 39

is 2-dimensional because we have two variables, x and y. There is a point (x, y) in the coordinate plane that satisfies both equations simultaneously. Solving such systems can be done using methods like substitution, elimination, or (as in our case) Cramer's Rule, which uses determinants to find a unique intersection point if the system is consistent and the coefficient matrix's determinant is non-zero.

Coefficient Matrix

The coefficient matrix in a system of linear equations is a matrix derived from the coefficients of the variables in the equations. It serves as a foundational tool in using Cramer's Rule. For the example given:

• From the equation 10x - 17y = 21
• From the equation 20x - 31y = 39

The coefficient matrix A is:\[A = \begin{pmatrix} 10 & -17 \ 20 & -31 \end{pmatrix}\]This matrix contains only the coefficients of x and y, excluding the constants. It organizes the system compactly and allows us to apply matrix operations, such as calculating the determinant or transforming it to solve for the variables using different mathematical techniques, including Cramer's Rule. A non-zero determinant of this matrix implies that the system has a unique solution, which was confirmed by solving the equations in this manner.