Problem 17 - Elementary and Intermediate Algebra

Question

Solve using Cramer's rule. $$ \begin{aligned} &5 x-4 y=-3\\\ &7 x+2 y=6 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{ 9 }{ 19 } \) and \( y = \frac{ 51 }{ 38 } \).

Step 1: Write the system of equations in matrix form

Convert the given system of equations into matrix form. The system of equations is: \[ \begin{cases} 5x - 4y = -3 \ 7x + 2y = 6 \end{cases} \] This can be represented in matrix form as: \[ A \mathbf{x} = \mathbf{b} \] where \[ A = \begin{pmatrix} 5 & -4 \ 7 & 2 \end{pmatrix}, \mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix}, \text{ and } \mathbf{b} = \begin{pmatrix} -3 \ 6 \end{pmatrix} \]

Step 2: Find the determinant of the coefficient matrix A

The determinant of matrix \( A \) is calculated as follows: \[ \det(A) = \begin{vmatrix} 5 & -4 \ 7 & 2 \end{vmatrix} = (5)(2) - (-4)(7) = 10 + 28 = 38 \]

Step 3: Find the determinant of matrix \( A_x \)

Replace the first column of matrix \( A \) with the constants from the matrix \( \mathbf{b} \) to form matrix \( A_x \): \[ A_x = \begin{pmatrix} -3 & -4 \ 6 & 2 \end{pmatrix} \] The determinant of matrix \( A_x \) is: \[ \det(A_x) = \begin{vmatrix} -3 & -4 \ 6 & 2 \end{vmatrix} = (-3)(2) - (-4)(6) = -6 + 24 = 18 \]

Step 4: Find the determinant of matrix \( A_y \)

Replace the second column of matrix \( A \) with the constants from the matrix \( \mathbf{b} \) to form matrix \( A_y \): \[ A_y = \begin{pmatrix} 5 & -3 \ 7 & 6 \end{pmatrix} \] The determinant of matrix \( A_y \) is: \[ \det(A_y) = \begin{vmatrix} 5 & -3 \ 7 & 6 \end{pmatrix} = (5)(6) - (-3)(7) = 30 + 21 = 51 \]

Step 5: Calculate \( x \) using Cramer's rule

Using Cramer's rule, the variable \( x \) is calculated as follows: \[ x = \frac{ \det(A_x) }{ \det(A) } = \frac{ 18 }{ 38 } = \frac{ 9 }{ 19 } \]

Step 6: Calculate \( y \) using Cramer's rule

Using Cramer's rule, the variable \( y \) is calculated as follows: \[ y = \frac{ \det(A_y) }{ \det(A) } = \frac{ 51 }{ 38 } \]

Key Concepts

system of linear equations

A system of linear equations is a set of two or more linear equations with the same variables. For example, in this exercise, we have two equations: \(5x - 4y = -3\) and \(7x + 2y = 6\).
The goal is to find the values of the variables \(x\) and \(y\) that satisfy both equations.
Solving these systems can be done using various methods, and one powerful method is Cramer's rule.

determinants

Determinants are special numbers calculated from a square matrix. They provide key properties of the matrix and help in solving systems of linear equations.
To find a determinant for a 2x2 matrix: \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( \text{det}(A) = ad - bc \).
In our problem, the coefficient matrix is \( A = \begin{pmatrix} 5 & -4 \ 7 & 2 \end{pmatrix} \) and its determinant is \( \text{det}(A) = (5)(2) - (-4)(7) = 38 \).
Determinants are crucial for applying Cramer's rule.

matrix form

Linear equations can be simplified using matrices.
The system of equations \(5x - 4y = -3\) and \(7x + 2y = 6\) can be written in matrix form as:
\ A\mathbf{x} = \mathbf{b} \ where \( A = \begin{pmatrix} 5 & -4 \ 7 & 2 \end{pmatrix} \), \( \mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix} \), and \( \mathbf{b} = \begin{pmatrix} -3 \ 6 \end{pmatrix} \).
This matrix form simplifies operations and is essential for using methods like Cramer's rule.

Cramer's rule

Cramer's rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns using determinants.
Here's how it works for our example:
1. Find the determinant of the coefficient matrix \(A\).
2. Create matrices \(A_x\) and \(A_y\) by replacing the respective columns of \(A\) with the constants from the right-hand side of the equations, \(\mathbf{b}\).
3. Calculate the determinants of \(A_x\) and \(A_y\).
4. Solve for the variables \(x\) and \(y\) using: \ \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \) and \ \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \)
In our problem, we calculated \( x = \frac{18}{38} = \frac{9}{19} \) and \( y = \frac{51}{38} \).
Cramer's rule is straightforward but powerful, making it easy to solve linear systems using matrices and determinants.