Problem 21 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 2 x+y=14 \\ 3 x-y=1 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \(x = 3\) and \(y = 8\).

Step 1: Writing the System of Equations

First, let's clearly define the system of linear equations given: 1. \(2x + y = 14\)2. \(3x - y = 1\)

Step 2: Expressing in Matrix Form

Next, we'll express the system in matrix form: \[\begin{pmatrix}2 & 1 \3 & -1\end{pmatrix}\begin{pmatrix}x \y\end{pmatrix} = \begin{pmatrix} 14 \ 1 \end{pmatrix}\]Where \(A\) is the coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constants matrix.

Step 3: Determinant of the Coefficient Matrix

Calculate the determinant of the coefficient matrix \(A\), which is required for Cramer's Rule:\[\text{det}(A) = \begin{vmatrix} 2 & 1 \ 3 & -1 \end{vmatrix} = (2)(-1) - (1)(3) = -2 - 3 = -5\]

Step 4: Determinant of Matrices for Individual Variables

To find \(x\), replace the first column of \(A\) with the constant matrix \(B\):\[A_x = \begin{pmatrix} 14 & 1 \ 1 & -1 \end{pmatrix}, \quad \text{det}(A_x) = \begin{vmatrix} 14 & 1 \ 1 & -1 \end{vmatrix} = (14)(-1) - (1)(1) = -14 - 1 = -15\]Similarly, to find \(y\), replace the second column of \(A\) with \(B\):\[A_y = \begin{pmatrix} 2 & 14 \ 3 & 1 \end{pmatrix}, \quad \text{det}(A_y) = \begin{vmatrix} 2 & 14 \ 3 & 1 \end{vmatrix} = (2)(1) - (14)(3) = 2 - 42 = -40\]

Step 5: Apply Cramer's Rule

Use Cramer's Rule to solve for \(x\) and \(y\):\[x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-15}{-5} = 3\]\[y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-40}{-5} = 8\]

Step 6: Conclusion

The solution set for the system of equations obtained by Cramer's rule is \(x = 3\) and \(y = 8\).

Key Concepts

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. These systems can have a single unique solution, no solution, or infinitely many solutions. In a simple sense, each equation represents a line, and the solution is the point where these lines intersect.
For example, consider the system:

• Equation 1: \(2x + y = 14 \)
• Equation 2: \(3x - y = 1\)

To solve these, we want to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Cramer's Rule is a convenient method to find the solution, especially for systems where the number of equations matches the number of variables.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It's a useful tool in linear algebra for solving systems of linear equations, among other applications. Only square matrices (same number of rows and columns) have determinants.

To find the determinant of a matrix \(A\) of size 2x2, use:

• Given: \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \)
• Determinant, \( \text{det}(A) = ad - bc \)

For the coefficient matrix \( \begin{pmatrix} 2 & 1 \ 3 & -1 \end{pmatrix} \), the determinant is \( (2)(-1) - (1)(3) = -2 - 3 = -5 \).
This value is crucial in Cramer's Rule, as it helps determine if the system has a unique solution (if the determinant is not zero).

Matrix Form

Converting a system of linear equations into matrix form is a structured approach that can simplify solving them.
In this form, the system of equations is represented using matrices:

• Coefficient matrix (\(A\)): contains the coefficients of the variables.
• Variable matrix (\(X\)): contains the variables themselves.
• Constants matrix (\(B\)): contains the results from each equation.

For the given example, we express the system as:\[\begin{pmatrix} 2 & 1 \ 3 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 14 \ 1 \end{pmatrix}\]This makes it easier to apply methods like matrix inverses or Cramer's Rule. It provides a visual and mathematical representation that's easier to manipulate than individual equations.