Problem 33 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 6 x-5 y=1 \\ 4 x+7 y=2 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is \( \left( \frac{17}{62}, \frac{4}{31} \right) \).

Step 1: Write the system in matrix form

Rewrite the system of equations in the form of a matrix equation. The equations given are: \(6x - 5y = 1\) and \(4x + 7y = 2\). This translates to the matrix equation \(AX = B\) where: \[ A = \begin{bmatrix} 6 & -5 \ 4 & 7 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad B = \begin{bmatrix} 1 \ 2 \end{bmatrix} \]

Step 2: Calculate the determinant of matrix A

The determinant of matrix \(A\), denoted as \(\det(A)\), is calculated as follows: \[ \det(A) = \begin{vmatrix} 6 & -5 \ 4 & 7 \end{vmatrix} = (6)(7) - (4)(-5) = 42 + 20 = 62 \]

Step 3: Compute determinants for matrices A_x and A_y

For Cramer's Rule, create matrices \(A_x\) and \(A_y\) by replacing the corresponding columns of \(A\) with \(B\): - Matrix \(A_x\) is \(A\) with the first column replaced by \(B\): \[ A_x = \begin{bmatrix} 1 & -5 \ 2 & 7 \end{bmatrix} \] - Matrix \(A_y\) is \(A\) with the second column replaced by \(B\): \[ A_y = \begin{bmatrix} 6 & 1 \ 4 & 2 \end{bmatrix} \]

Step 4: Calculate determinants of A_x and A_y

Find the determinants of these modified matrices:- \( \det(A_x) = \begin{vmatrix} 1 & -5 \ 2 & 7 \end{vmatrix} = (1)(7) - (2)(-5) = 7 + 10 = 17 \) - \( \det(A_y) = \begin{vmatrix} 6 & 1 \ 4 & 2 \end{vmatrix} = (6)(2) - (4)(1) = 12 - 4 = 8 \)

Step 5: Apply Cramer's Rule to solve for x and y

Cramer's Rule states that: \[x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}\]Substitute the known determinants:\[x = \frac{17}{62}, \quad y = \frac{8}{62}\]Simplify the fractions to:\[x = \frac{17}{62}, \quad y = \frac{4}{31}\]

Step 6: State the solution set

The solution set for the system of equations is given by the values of \(x\) and \(y\) obtained using Cramer's Rule:\[ (x, y) = \left( \frac{17}{62}, \frac{4}{31} \right) \]

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It plays a vital role in matrix algebra, particularly in solving systems of equations. For a 2x2 matrix \(A = \begin{bmatrix}a & b \ c & d\end{bmatrix}\), the determinant, denoted as \(\det(A)\), is calculated as \[ ad - bc \]. This equation might seem simple, but the determinant gives important information about the matrix, like whether it is invertible. If the determinant is zero, the matrix does not have an inverse, and you cannot use methods like Cramer's Rule to find a unique solution to a system of equations. For our specific example, the determinant \(\det(A) = 62\) tells us that the matrix is invertible, hence allowing us to proceed with solving the system using Cramer's Rule.

Matrix Equation

A matrix equation represents a system of linear equations using matrices. In mathematical terms, the format is given by \(AX = B\). Here, \(A\) is the coefficient matrix containing the coefficients of the variables, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants. For example, in the system of equations \(6x - 5y = 1\) and \(4x + 7y = 2\), we can express this as the matrix equation:

• \(A = \begin{bmatrix} 6 & -5 \ 4 & 7 \end{bmatrix}\)
• \(X = \begin{bmatrix} x \ y \end{bmatrix}\)
• \(B = \begin{bmatrix} 1 \ 2 \end{bmatrix}\)

The matrix equation efficiently organizes the system and is essential for applying techniques like Cramer's Rule. This method transforms solving systems into managing matrix operations.

System of Equations

A system of equations consists of two or more equations that have common variables. Solving the system means finding the values of these variables that satisfy all the equations simultaneously. The classic methods for solution include substitution, elimination, and using matrix algebra. Cramer's Rule stands out because it uses determinants to find the solution of the system. Consider the equations \(6x - 5y = 1\) and \(4x + 7y = 2\). These represent our system of equations. To use Cramer's Rule, the determinant of the coefficients matrix must be non-zero, which ensures a unique solution. Using determinants, you solve for each variable by replacing columns of the original matrix with the constant matrix.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with operations on matrices. It forms the basis for solving complex problems involving multiple equations and variables, like in engineering and computer graphics. It includes operations like addition, subtraction, multiplication, and finding inverses of matrices. One powerful application is solving systems of equations. Using matrix algebra methods, such as Cramer's Rule, we calculate the determinants to find solutions. For a given matrix \(A\) related to our system:

• Calculate its determinant \(\det(A)\)
• Form secondary matrices like \(A_x\) and \(A_y\) by swapping columns with the constant matrix \(B\)
• Calculate determinants for these matrices and apply Cramer's formula to solve for variables.

Matrix algebra not only provides solutions but also reveals properties about systems, like uniqueness and consistency of solutions.