Problem 17 - Technical Mathematics with Calculus

Question

Solve by determinants. $$\begin{aligned} &x+5 y=11\\\ &3 x+2 y=7 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1\) and \(y = 2\).

Step 1: Write Down the System of Equations

The system of equations given is: \[ \begin{aligned} &x + 5y = 11, \ &3x + 2y = 7. \end{aligned} \] We are to solve for the variables \(x\) and \(y\) by using determinants.

Step 2: Identify the Coefficient Matrix and Constants Matrix

The coefficient matrix is given by: \[ A = \begin{bmatrix} 1 & 5 \ 3 & 2 \end{bmatrix} \] And the constants matrix (column matrix of the right-hand side of the equations) is: \[ B = \begin{bmatrix} 11 \ 7 \end{bmatrix} \]

Step 3: Compute the Determinant of Matrix A

To find the solution, we need the determinant of matrix \(A\): \[ \text{det}(A) = (1)(2) - (5)(3) = 2 - 15 = -13. \] The determinant \(\text{det}(A)\) must be non-zero to have a unique solution.

Step 4: Compute Determinants for \(x\) and \(y\)

Replace the first column of \(A\) with matrix \(B\) for \(x\), and find the determinant: \[ A_x = \begin{bmatrix} 11 & 5 \ 7 & 2 \end{bmatrix}, \quad \text{det}(A_x) = (11)(2) - (5)(7) = 22 - 35 = -13. \] Replace the second column of \(A\) with matrix \(B\) for \(y\), and find the determinant: \[ A_y = \begin{bmatrix} 1 & 11 \ 3 & 7 \end{bmatrix}, \quad \text{det}(A_y) = (1)(7) - (11)(3) = 7 - 33 = -26. \]

Step 5: Solve for \(x\) and \(y\)

Using Cramer's Rule, solve for \(x\) and \(y\): \[ x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}. \] Substituting the calculated determinants: \[ x = \frac{-13}{-13} = 1, \quad y = \frac{-26}{-13} = 2. \]

Key Concepts

Cramer's Rule

Cramer's Rule is a theorem in algebra that provides an explicit solution to a system of linear equations with as many equations as unknowns, given that the system has a unique solution. The rule uses determinants of matrices to find the solution for each variable in the system. According to Cramer's Rule, if you have a system of linear equations represented in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constants matrix, the solution for each variable x_i can be found by substituting the i^th column of A with B and dividing the resulting determinant by the determinant of the original coefficient matrix A.

It is crucial that the determinant of matrix A (denoted as det(A)) is non-zero, as a zero determinant indicates that the system of equations does not have a unique solution. If the determinant is non-zero, the unique solution for each variable x_i is given by:
\[ x_i = \frac{\text{det}(A_i)}{\text{det}(A)} \]
where A_i is the matrix formed by replacing the i^th column of A with the constants matrix B.

Coefficient Matrix

In the context of solving systems of linear equations, the coefficient matrix is a square matrix that contains the coefficients of the variables in the system. Each equation in the system contributes one row to the matrix, and each variable contributes one column. For instance, if we are working with the equations:
\[ x + 5y = 11 \]
\[ 3x + 2y = 7 \]
The coefficient matrix would be:
\[ A = \begin{bmatrix} 1 & 5 \ 3 & 2 \end{bmatrix} \]
This matrix is critical in Cramer's Rule as it is used to calculate the determinants needed to solve for the variables. The integrity of the coefficient matrix is essential, as the existence of a unique solution depends on its determinant not being zero.

Determinant of a Matrix

The determinant of a matrix is a special scalar value that can be calculated from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible, and the volume scaling factor of the linear transformation associated with the matrix. The determinant of a 2x2 matrix:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
is computed as:
\[\text{det}(A) = ad - bc\]
The calculation of the determinant is a key step in Cramer's Rule for solving a system of equations. In our example, the determinant of the coefficient matrix A is:
\[\text{det}(A) = (1)(2) - (5)(3) = 2 - 15 = -13\]
A non-zero determinant is a signal that the system has a unique solution, which is the precondition for applying Cramer's Rule effectively.

Cramer's Rule Example

Let's apply Cramer's Rule to a concrete example:
The system of equations to solve is given by:
\[\begin{aligned} x + 5y &= 11, \ 3x + 2y &= 7. \end{aligned}\]
Following the steps of Cramer's Rule, we first determine the coefficient matrix and calculate its determinant. Next, we compute the determinants for each variable by replacing the corresponding column with the constants matrix. For our given example:
\[A_x = \begin{bmatrix} 11 & 5 \ 7 & 2 \end{bmatrix},\quad A_y = \begin{bmatrix} 1 & 11 \ 3 & 7 \end{bmatrix}\]
The determinants are then:
\[\text{det}(A_x) = 22 - 35 = -13\]
\[\text{det}(A_y) = 7 - 33 = -26\]
Finally, we solve for x and y using Cramer's Rule:
\[x = \frac{-13}{-13} = 1,\quad y = \frac{-26}{-13} = 2\]
Thus, the solution to the system of equations using Cramer's Rule is x = 1 and y = 2. This method provides a systematic and straightforward approach to solving systems of equations when the determinant of the coefficient matrix is non-zero.